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A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model

A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a... Article A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model Fabio Silva Botelho Department of Mathematics, Federal University of Santa Catarina, UFSC, Florianópolis 88040-900, Brazil; fabio.botelho@ufsc.br Abstract: In the first part of this article, we present a new proof for Korn’s inequality in an n- dimensional context. The results are based on standard tools of real and functional analysis. For the final result, the standard Poincaré inequality plays a fundamental role. In the second text part, we develop a global existence result for a non-linear model of plates. We address a rather general type of boundary conditions and the novelty here is the more relaxed restrictions concerning the external load magnitude. Keywords: Korn’s inequality; global existence result; non-linear plate model MSC: 35Q74; 35J58 1. Introduction In this article, we present a proof for Korn’s inequality in R . The results are based on standard tools of functional analysis and on the Sobolev spaces theory. We emphasize that such a proof is relatively simple and easy to follow since it is established in a very transparent and clear fashion. About the references, we highlight that related results in a three-dimensional context may be found in [1]. Other important classical results on Korn’s inequality and concerning applications to models in elasticity may be found in [2–4]. Citation: Botelho, F.S. A Note on Remark 1. Generically, throughout the text we denote Korn’s Inequality in an N-Dimensional Context and a Global 1/2 2 2 Existence Result for a Non-Linear kuk = juj dx , 8u 2 L (W), 0,2,W Plate Model. AppliedMath 2023, 3, 406–416. https://doi.org/10.3390/ and appliedmath3020021 1/2 2 2 n kuk = ku k , 8u = (u , . . . , u ) 2 L (W;R ). Received: 15 March 2023 0,2,W å j 1 0,2,W j=1 Revised: 12 April 2023 Accepted: 17 April 2023 Moreover, Published: 2 May 2023 1/2 2 2 1,2 kuk = kuk + ku k , 8u 2 W (W), 1,2,W å 0,2,W j 0,2,W j=1 Copyright: © 2023 by the authors. where we shall also refer throughout the text to the well-known corresponding analogous norm for Licensee MDPI, Basel, Switzerland. 1,2 n u 2 W (W;R ). This article is an open access article distributed under the terms and At this point, we first introduce the following definition. conditions of the Creative Commons Attribution (CC BY) license (https:// n 1 Definition 1. Let W  R be an open, bounded set. We say that ¶W is C if such a manifold is creativecommons.org/licenses/by/ oriented and for each x 2 ¶W, denoting x ˆ = (x , ..., x ) for a local coordinate system compatible 0 1 n1 4.0/). AppliedMath 2023, 3, 406–416. https://doi.org/10.3390/appliedmath3020021 https://www.mdpi.com/journal/appliedmath AppliedMath 2023, 3 407 with the manifold ¶W orientation, there exist r > 0 and a function f (x , ..., x ) = f (x ˆ) such 1 n1 that W = W\ B (x ) = fx 2 B (x ) j x  f (x , ..., x )g. r 0 r 0 n 1 n1 Moreover, f (x ˆ) is a Lipschitz continuous function, so that j f (x ˆ) f (y ˆ)j  C jx ˆ y ˆj , on its domain, 1 2 for some C > 0. Finally, we assume n1 ¶ f (x ˆ) ¶x k=1 1,2 is classically defined, almost everywhere also on its concerning domain, so that f 2 W . Remark 2. This mentioned set W is of a Lipschitzian type, so that we may refer to such a kind of sets as domains with a Lipschitzian boundary, or simply as Lipschitzian sets. At this point, we recall the following result found in [5], at page 222 in its Chapter 11. n 1 Theorem 1. Assume W  R is an open bounded set, and that ¶W is C . Let 1  p < ¥, and let V be a bounded open set such that W  V. Then there exists a bounded linear operator 1, p 1, p n E : W (W) ! W (R ), 1, p such that for each u 2 W (W) we have: 1. Eu = u, a.e. in W,; 2. Eu has support in V; 3. kEuk n  Ckuk , where the constant depends only on p, W, and V. 1, p,R 1, p,W Remark 3. Considering the proof of such a result, the constant C > 0 may be also such that 1,2 n ke (Eu)k  C(ke (u)k +kuk ), 8u 2 W (W;R ), 8i, j 2 f1, . . . , ng, i j 0,2,V i j 0,2,W 0,2,W 1,2 n 2 nn for the operator e : W (W;R ) ! L (W;R ) specified in the next theorem. Finally, as the meaning is clear, we may simply denote Eu = u. 2. The Main Results, the Korn Inequalities Our main result is summarized by the following theorem. n 1 Theorem 2. Let W  R be an open, bounded and connected set with a C (Lipschitzian) bound- ary ¶W. 1,2 n 2 nn Define e : W (W;R ) ! L (W;R ) by e(u) = fe (u)g i j where e (u) = (u + u ), 8i, j 2 f1, . . . , ng, i j i,j j,i and where generically, we denote ¶u u = , 8i, j 2 f1, , ng. i,j ¶x j AppliedMath 2023, 3 408 Define also, 1/2 n n ke(u)k = ke k . 0,2,W å å i j(u) 0,2,W i=1 j=1 + n 0 Let L 2 R be such V = [L, L] is also such that W  V . Under such hypotheses, there exists C(W, L) 2 R such that 1,2 n kuk  C(W, L)(kuk +ke(u)k ), 8u 2 W (W;R ). (1) 1,2,W 0,2,W 0,2,W Proof. Suppose, to obtain contradiction, that the concerning claim does not hold. Thus, we are assuming that there is no positive real constant C = C(W, L) such that (1) is valid. In particular, k = 1 2 N is not such a constant C, so that there exists a function 1,2 n u 2 W (W;R ) such that ku k > 1 (ku k +ke(u )k ). 1 1,2,W 1 0,2,W 1 0,2,W Similarly, k = 2 2 N is not such a constant C, so that there exists a function u 2 1,2 n W (W;R ) such that ku k > 2 (ku k +ke(u )k ). 2 1,2,W 2 0,2,W 2 0,2,W Hence, since no k 2 N is such a constant C, reasoning inductively, for each k 2 N there 1,2 n exists a function u 2 W (W;R ) such that ku k > k(ku k +ke(u )k ). k 1,2,W k 0,2,W k 0,2,W In particular, defining v = ku k k 1,2,W we obtain kv k = 1 > k(kv k +ke(v )k ), k 1,2,W k 0,2,W k 0,2,W so that (kv k +ke(v )k ) < , 8k 2 N. k 0,2,W k 0,2,W From this we obtain kv k < , k 0,2,W and ke (v )k < , 8k 2 N, i j k 0,2,W so that kv k ! 0, as k ! ¥, k 0,2,W and ke (v )k ! 0, as k ! ¥. 0,2,W i j k In particular, k(v ) k ! 0, 8j 2 f1, . . . , ng. k j,j 0,2,W At this point, we recall the following identity in the distributional sense, found in [3], page 12: ¶ (¶ v ) = ¶ e (v) + ¶ e (v) ¶ e (v), 8i, j, l 2 f1, . . . , ng. (2) j l i j il l i j i jl Fix j 2 f1, . . . , ng and observe that k(v ) k  Ck(v ) k , k j 1,2,V k j 1,2,W AppliedMath 2023, 3 409 so that C 1 , 8k 2 N. k(v ) k k(v ) k k j 1,2,V k j 1,2,W Hence, k(v ) k j 1,2,W n o = sup hr(v ) ,rji +h(v ) , ji : kjk  1 2 2 k j k j 1,2,W L (W) L (W) j2C (W) * !+ (v ) k j = r(v ) ,r k j k(v ) k k j 1,2,W L (W) * !+ (v ) k j + (v ) , k(v ) k k j 1,2,W L (W) 0 1 * !+ * !+ (v ) (v ) k j k j @ A C r(v ) ,r + (v ) , k j k j k(v ) k k(v ) k 1,2,V 1,2,V k j k j 2 2 L (V) L (V) n o = C sup hr(v ) ,rji 2 +h(v ) , ji 2 : kjk  1 . (3) 1,2,V k j L (V) k j L (V) j2C (V) Here, we recall that C > 0 is the constant concerning the extension Theorem 1. From such results and (2), we have that n o sup hr(v ) ,rji +h(v ) , ji : kjk  1 2 2 k j k j 1,2,W L (W) L (W) j2C (W) n o C sup hr(v ) ,rji 2 +h(v ) , ji 2 : kjk  1 1,2,V k j L (V) k j L (V) j2C (V) = C sup he (v ), j i +he (v ), j i 2 2 jl k ,l jl k ,l L (V) L (V) j2C (V) he (v ), j i +h(v ) , ji , : kjk  1 . (4) 2 2 ll k ,j k j 1,2,V L (V) L (V) Therefore, k(v ) k k j 1,2 W (W) ( ) = sup fhr(v ) ,rji +h(v ) , ji : kjk  1g 2 2 k j k j 1,2,W L (W) L (W) j2C (W) n o C ke (v )k +ke (v )k +k(v ) k å jl k 0,2,V ll k 0,2,V k j 0,2,V l=1 n o C ke (v )k +ke (v )k +k(v ) k 1 å jl k 0,2,W ll k 0,2,W k j 0,2,W l=1 < , (5) for appropriate C > 0 and C > 0. 1 2 Summarizing, k(v ) k < , 8k 2 N. 1,2 k j (W (W)) From this we obtain kv k ! 0, as k ! ¥, k 1,2,W AppliedMath 2023, 3 410 which contradicts kv k = 1, 8k 2 N. k 1,2,W The proof is complete. n 1 Corollary 1. Let W  R be an open, bounded and connected set with a C boundary ¶W. Define 1,2 n 2 nn e : W (W;R ) ! L (W;R ) by e(u) = fe (u)g i j where e (u) = (u + u ), 8i, j 2 f1, . . . , ng. i j i,j j,i Define also, 1/2 n n ke(u)k = ke k . 0,2,W å å i j(u) 0,2,W i=1 j=1 + n 0 Let L 2 R be such V = [L, L] is also such that W  V . Moreover, define 1,2 n H = fu 2 W (W;R ) : u = 0, on G g, 0 0 where G  ¶W is a measurable set such that the Lebesgue measure m n1(G ) > 0. 0 0 Assume also G is such that for each j 2 f1, , ng and each x = (x , , x ) 2 W there 0 1 n exists x = ((x ) , , (x ) ) 2 G such that 0 0 0 n 0 (x ) = x , 8l 6= j, l 2 f1, , ng, l l and the line A  W x ,x where A = f(x , , (1 t)(x ) + tx , , x ) : t 2 [0, 1]g. x ,x 0 n 1 j j Under such hypotheses, there exists C(W, L) 2 R such that kuk  C(W, L) ke(u)k , 8u 2 H . 1,2,W 0,2,W 0 Proof. Suppose, to obtain contradiction, that the concerning claim does not hold. Hence, for each k 2 N there exists u 2 H such that ku k > k ke(u )k . k 1,2,W k 0,2,W In particular, defining v = ku k k 1,2,W similarly to the proof of the last theorem, we may obtain k(v ) k ! 0, as k ! ¥, 8j 2 f1, . . . , ng. k j,j 0,2,W From this, the hypotheses on G and from the standard Poincaré inequality proof we obtain k(v ) k ! 0, as k ! ¥, 8j 2 f1, . . . , ng. j 0,2,W Thus, also similarly as in the proof of the last theorem, we may infer that kv k ! 0, as k ! ¥, k 1,2,W AppliedMath 2023, 3 411 which contradicts kv k = 1, 8k 2 N. k 1,2,W The proof is complete. 3. An Existence Result for a Non-Linear Model of Plates In the present section, as an application of the results on Korn’s inequalities presented in the previous sections, we develop a new global existence proof for a Kirchhoff–Love thin plate model. Previous results on the existence of mathematical elasticity and related models may be found in [2–4]. At this point we start to describe the primal formulation. Let W  R be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of W, which is assumed to be regular (Lipschitzian), is denoted by ¶W. The vectorial basis related to the cartesian system fx , x , x g is denoted 2 3 by (a , a ), where a = 1, 2 (in general, Greek indices stand for 1 or 2), and where a is the a 3 3 vector normal to W, whereas a and a are orthogonal vectors parallel to W. Furthermore, n is the outward normal to the plate surface. The displacements will be denoted by u ˆ = fu ˆ , u ˆ g = u ˆ a + u ˆ a . a a a 3 3 3 The Kirchhoff–Love relations are u ˆ (x , x , x ) = u (x , x ) x w(x , x ) a a ,a 1 2 3 1 2 3 1 2 and u ˆ (x , x , x ) = w(x , x ). (6) 3 1 2 3 1 2 Here, h/2  x  h/2 so that we have u = (u , w) 2 U where 3 a 1,2 2 2,2 U = (u , w) 2 W (W;R ) W (W), ¶w u = w = = 0 on ¶W ¶n 1,2 2 2,2 = W (W;R ) W (W). 0 0 It is worth emphasizing that the boundary conditions here specified refer to a clamped plate. 2 22 We define the operator L : U ! Y Y, where Y = Y = L (W;R ), by L(u) = fg(u), k(u)g, u + u w w a,b b,a ,a ,b g (u) = + , ab 2 2 k (u) = w . ab ,ab The constitutive relations are given by N (u) = H g (u), (7) ab ablm lm M (u) = h k (u), (8) ab ablm lm where f H g and fh = H g, are symmetric positive definite fourth-order ablm ablm ablm 1 1 tensors. From now on, we denote f H g = f H g and fh g = fh g . ablm ablm ablm ablm AppliedMath 2023, 3 412 Furthermore, fN g denote the membrane force tensor and f M g the moment one. ab ab The plate stored energy, represented by (G L) : U ! R, is expressed by Z Z 1 1 (G L)(u) = N (u)g (u) dx + M (u)k (u) dx (9) ab ab ab ab 2 2 W W and the external work, represented by F : U ! R, is given by F(u) = hw, Pi 2 +hu , P i 2 , (10) a a L (W) L (W) where P, P , P 2 L (W) are external loads in the directions a , a , and a , respectively. The 1 2 3 1 2 potential energy, denoted by J : U ! R, is expressed by J(u) = (G L)(u) F(u) Finally, we also emphasize from now on, as their meaning are clear, we may denote 2 2 22 2 L (W) and L (W;R ) simply by L , and the respective norms bykk . Moreover, deriva- tives are always understood in the distributional sense, 0 may denote the zero vector in appropriate Banach spaces, and the following and relating notations are used: ¶w w = , ,a ¶x ¶ w w = , ,ab ¶x ¶x a b ¶u u = , a,b ¶x ¶N ab N = , ab,1 ¶x and ¶N ab N = . ab,2 ¶x 4. On the Existence of a Global Minimizer At this point, we present an existence result concerning the Kirchhoff–Love plate model. We start with the following two remarks. ¥ 2 Remark 4. Let fP g 2 L (W;R ). We may easily obtain by appropriate Lebesgue integration fT g symmetric and such that ab T = P , in W. ab,b Indeed, extending fP g to zero outside W if necessary, we may set T (x, y) = P (x, y) dx, 11 1 T (x, y) = P (x, x) dx, 22 2 and ˜ ˜ T (x, y) = T (x, y) = 0, in W. 12 21 Thus, we may choose a C > 0 sufficiently big, such that fT g = fT + Cd g ab ab ab AppliedMath 2023, 3 413 is positive definite in W, so that T = T = P , ab,b ab,b a where fd g ab is the Kronecker delta. Therefore, for the kind of boundary conditions of the next theorem, we do not have any restriction for the fP g norm. In summary, the next result is new and it is really a step forward concerning the previous one in Ciarlet [3]. We emphasize that this result and its proof through such a tensor fT g are new, ab even though the final part of the proof is established through a standard procedure in the calculus of variations. Finally, more details on the Sobolev spaces involved may be found in [5–8]. Related duality principles are addressed in [5,7,9]. At this point, we present the main theorem in this section. Theorem 3. Let W  R be an open, bounded, connected set with a Lipschitzian boundary denoted by ¶W = G. Suppose (G L) : U ! R is defined by G(Lu) = G (g(u)) + G (k(u)), 8u 2 U, 1 2 where G (gu) = H g (u)g (u) dx, ablm ab lm and G (ku) = h k (u)k (u) dx, ablm ab lm where L(u) = (g(u), k(u)) = (fg (u)g,fk (u)g), ab ab u + u w w a,b b,a ,a ,b g (u) = + , ab 2 2 k (u) = w , ab ,ab and where J(u) = W(g(u), k(u))hP , u i 2 a a L (W) hw, Pi 2 hP , u i 2 L (W) a L (G ) hP , wi , (11) L (G ) where, 1,2 2 2,2 U = fu = (u , w) = (u , u , w) 2 W (W;R ) W (W) : a 1 2 ¶w u = w = = 0, on G g, (12) ¶n where ¶W = G [ G and the Lebesgue measures 0 t m (G \ G ) = 0, G 0 t and m (G ) > 0. G 0 AppliedMath 2023, 3 414 We also define F (u) = hw, Pi hu , P i hP , u i 2 2 2 1 a a a L (W) L (W) a L (G ) t 2 hP , wi 2 +h# , u i 2 L (G ) a L (G ) t t hu, fi 2 +h# , u i 2 L a L (G ) hu, f i 2 hu , P i 2 +h# , u i 2 , (13) a a a 1 a L L (W) L (G ) where hu, f i = hu, fi hu , P i , 2 2 2 1 a a L L L (W) # > 0, 8a 2 f1, 2g and ¥ 3 f = (P , P) 2 L (W;R ). Let J : U ! R be defined by J(u) = G(Lu) + F (u), 8u 2 U. 22 Assume there exists fc g 2 R such that c > 0, 8a, b 2 f1, 2g and ab ab G (k(u))  c kw k , 8u 2 U. 2 ab ,ab Under such hypotheses, there exists u 2 U such that J(u ) = min J(u). u2U Proof. Observe that we may find T = f(T ) g such that a a b divT = T = P , a ab,b a and also such that fT g is positive, definite, and symmetric (please see Remark 4). ab Thus, defining u + u a,b b,a v (u) = + w w , (14) ,a ab ,b 2 2 we obtain J(u) = G (fv (u)g) + G (k(u))hu, fi 2 +h# , u i 2 2 a 1 ab a L L (G ) = G (fv (u)g) + G (k(u)) +hT , u i 2 hu, f i 2 +h# , u i 2 a a 1 ab 2 ab,b 1 a L (W) L L (G ) u + u a,b b,a = G (fv (u)g) + G (k(u)) T , 1 ab 2 ab L (W) +hT n , u i hu, f i +h# , u i 2 2 2 ab b a 1 a L (G ) L a L (G ) t t = G (fv (u)g) + G (k(u)) T , v (u) w w hu, f i +h# , u i 2 2 1 ab 2 ab ab ,a ,b 1 a L a L (G ) L (W) +hT n , u i a 2 ab b L (G ) 2 2 c kw k + T , w w hu, f i 2 +h# , u i 2 + G (fv (u)g) ab ,ab ab ,a ,b 1 a 1 ab 2 2 L a L (G ) L (W) t hT , v (u)i +hT n , u i . (15) 2 2 ab ab ab b a L (W) L (G ) From this, since fT g is positive definite, clearly J is bounded below. ab Let fu g 2 U be a minimizing sequence for J. Thus, there exists a 2 R such that lim J(u ) = inf J(u) = a . n!¥ u2U AppliedMath 2023, 3 415 From (15), there exists K > 0 such that k(w ) k < K ,8a, b 2 f1, 2g, n 2 N. n ,ab 2 1 2,2 Therefore, there exists w 2 W (W) such that, up to a subsequence not relabeled, (w ) * (w ) , weakly in L , ,ab 0 ,ab 8a, b 2 f1, 2g, as n ! ¥. Moreover, also up to a subsequence not relabeled, 2 4 (w ) ! (w ) , strongly in L and L , (16) n ,a 0 ,a 8a,2 f1, 2g, as n ! ¥. Furthermore, from (15), there exists K > 0 such that, k(v ) (u)k < K ,8a, b 2 f1, 2g, n 2 N, ab 2 2 and thus, from this, (14) and (16), we may infer that there exists K > 0 such that k(u ) + (u ) k < K ,8a, b 2 f1, 2g, n 2 N. n n a,b b,a 2 3 From this and Korn’s inequality, there exists K > 0 such that ku k  K , 8n 2 N. 1,2 2 n 4 W (W;R ) 1,2 2 Therefore, up to a subsequence not relabeled, there exists f(u ) g 2 W (W,R ), such that 0 a (u ) + (u ) * (u ) + (u ) , weakly in L , n a,b n b,a 0 a,b 0 b,a 8a, b 2 f1, 2g, as n ! ¥, and (u ) ! (u ) , strongly in L , n a 0 a 8a 2 f1, 2g, as n ! ¥. Moreover, the boundary conditions satisfied by the subsequences are also satisfied for w and u in a trace sense, so that 0 0 u = ((u ) , w ) 2 U. 0 0 a 0 From this, up to a subsequence not relabeled, we obtain g (u ) * g (u ), weakly in L , ab n ab 0 8a, b 2 f1, 2g, and k (u ) * k (u ), weakly in L , ab ab 0 8a, b 2 f1, 2g. Therefore, from the convexity of G in g and G in k, we obtain 1 2 inf J(u) = a u2U = lim inf J(u ) n!¥ J(u ). (17) Thus, J(u ) = min J(u). u2U AppliedMath 2023, 3 416 The proof is complete. 5. Conclusions In this article, we have developed a new proof for Korn’s inequality in a specific n-dimensional context. In the second text part, we present a global existence result for a non-linear model of plates. Both results represent some new advances concerning the present literature. In particular, the results for Korn’s inequality known so far are for a three-dimensional context such as in [1], for example, whereas we have here addressed a more general n-dimensional case. In a future research, we intend to address more general models, including the corre- sponding results for manifolds in R . Funding: This research received no external funding Conflicts of Interest: The author declares no conflict of interest. References 1. Lebedev, L.P.; Cloud, M.J. Korn’s Inequality. In Encyclopedia of Continuum Mechanics; Altenbach, H., Öchsner, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2020. [CrossRef] 2. Ciarlet, P. Mathematical Elasticity; Vol. I—Three Dimensional Elasticity; Elsevier: Amsterdam, The Netherlands, 1988. 3. Ciarlet, P. Mathematical Elasticity; Vol. II—Theory of Plates; Elsevier: Amsterdam, The Netherlands, 1997. 4. Ciarlet, P. Mathematical Elasticity; Vol. III—Theory of Shells; Elsevier: Amsterdam, The Netherlands, 2000. 5. Botelho, F.S. Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering; CRC Taylor and Francis: Uttar Pradesh, India, 2020. 6. Adams, R.A.; Fournier, J.F. Sobolev Spaces, 2nd ed.; Elsevier: New York, NY, USA, 2003. 7. Botelho, F.S. Functional Analysis and Applied Optimization in Banach Spaces; Springer: Cham, Switzerland, 2014. 8. Evans, L.C. Partial Differential Equations. In Graduate Studies in Mathematics; AMS: Providence, RI, USA, 1998. 9. Ekeland, I.; Temam, R. Convex Analysis and Variational Problems; Elsevier: Amsterdam, The Netherlands, 1976. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png AppliedMath Multidisciplinary Digital Publishing Institute

A Note on Korn&rsquo;s Inequality in an N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model

Botelho, Fabio Silva
AppliedMath , Volume 3 (2) – May 2, 2023
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Article A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model Fabio Silva Botelho Department of Mathematics, Federal University of Santa Catarina, UFSC, Florianópolis 88040-900, Brazil; fabio.botelho@ufsc.br Abstract: In the first part of this article, we present a new proof for Korn’s inequality in an n- dimensional context. The results are based on standard tools of real and functional analysis. For the final result, the standard Poincaré inequality plays a fundamental role. In the second text part, we develop a global existence result for a non-linear model of plates. We address a rather general type of boundary conditions and the novelty here is the more relaxed restrictions concerning the external load magnitude. Keywords: Korn’s inequality; global existence result; non-linear plate model MSC: 35Q74; 35J58 1. Introduction In this article, we present a proof for Korn’s inequality in R . The results are based on standard tools of functional analysis and on the Sobolev spaces theory. We emphasize that such a proof is relatively simple and easy to follow since it is established in a very transparent and clear fashion. About the references, we highlight that related results in a three-dimensional context may be found in [1]. Other important classical results on Korn’s inequality and concerning applications to models in elasticity may be found in [2–4]. Citation: Botelho, F.S. A Note on Remark 1. Generically, throughout the text we denote Korn’s Inequality in an N-Dimensional Context and a Global 1/2 2 2 Existence Result for a Non-Linear kuk = juj dx , 8u 2 L (W), 0,2,W Plate Model. AppliedMath 2023, 3, 406–416. https://doi.org/10.3390/ and appliedmath3020021 1/2 2 2 n kuk = ku k , 8u = (u , . . . , u ) 2 L (W;R ). Received: 15 March 2023 0,2,W å j 1 0,2,W j=1 Revised: 12 April 2023 Accepted: 17 April 2023 Moreover, Published: 2 May 2023 1/2 2 2 1,2 kuk = kuk + ku k , 8u 2 W (W), 1,2,W å 0,2,W j 0,2,W j=1 Copyright: © 2023 by the authors. where we shall also refer throughout the text to the well-known corresponding analogous norm for Licensee MDPI, Basel, Switzerland. 1,2 n u 2 W (W;R ). This article is an open access article distributed under the terms and At this point, we first introduce the following definition. conditions of the Creative Commons Attribution (CC BY) license (https:// n 1 Definition 1. Let W  R be an open, bounded set. We say that ¶W is C if such a manifold is creativecommons.org/licenses/by/ oriented and for each x 2 ¶W, denoting x ˆ = (x , ..., x ) for a local coordinate system compatible 0 1 n1 4.0/). AppliedMath 2023, 3, 406–416. https://doi.org/10.3390/appliedmath3020021 https://www.mdpi.com/journal/appliedmath AppliedMath 2023, 3 407 with the manifold ¶W orientation, there exist r > 0 and a function f (x , ..., x ) = f (x ˆ) such 1 n1 that W = W\ B (x ) = fx 2 B (x ) j x  f (x , ..., x )g. r 0 r 0 n 1 n1 Moreover, f (x ˆ) is a Lipschitz continuous function, so that j f (x ˆ) f (y ˆ)j  C jx ˆ y ˆj , on its domain, 1 2 for some C > 0. Finally, we assume n1 ¶ f (x ˆ) ¶x k=1 1,2 is classically defined, almost everywhere also on its concerning domain, so that f 2 W . Remark 2. This mentioned set W is of a Lipschitzian type, so that we may refer to such a kind of sets as domains with a Lipschitzian boundary, or simply as Lipschitzian sets. At this point, we recall the following result found in [5], at page 222 in its Chapter 11. n 1 Theorem 1. Assume W  R is an open bounded set, and that ¶W is C . Let 1  p < ¥, and let V be a bounded open set such that W  V. Then there exists a bounded linear operator 1, p 1, p n E : W (W) ! W (R ), 1, p such that for each u 2 W (W) we have: 1. Eu = u, a.e. in W,; 2. Eu has support in V; 3. kEuk n  Ckuk , where the constant depends only on p, W, and V. 1, p,R 1, p,W Remark 3. Considering the proof of such a result, the constant C > 0 may be also such that 1,2 n ke (Eu)k  C(ke (u)k +kuk ), 8u 2 W (W;R ), 8i, j 2 f1, . . . , ng, i j 0,2,V i j 0,2,W 0,2,W 1,2 n 2 nn for the operator e : W (W;R ) ! L (W;R ) specified in the next theorem. Finally, as the meaning is clear, we may simply denote Eu = u. 2. The Main Results, the Korn Inequalities Our main result is summarized by the following theorem. n 1 Theorem 2. Let W  R be an open, bounded and connected set with a C (Lipschitzian) bound- ary ¶W. 1,2 n 2 nn Define e : W (W;R ) ! L (W;R ) by e(u) = fe (u)g i j where e (u) = (u + u ), 8i, j 2 f1, . . . , ng, i j i,j j,i and where generically, we denote ¶u u = , 8i, j 2 f1, , ng. i,j ¶x j AppliedMath 2023, 3 408 Define also, 1/2 n n ke(u)k = ke k . 0,2,W å å i j(u) 0,2,W i=1 j=1 + n 0 Let L 2 R be such V = [L, L] is also such that W  V . Under such hypotheses, there exists C(W, L) 2 R such that 1,2 n kuk  C(W, L)(kuk +ke(u)k ), 8u 2 W (W;R ). (1) 1,2,W 0,2,W 0,2,W Proof. Suppose, to obtain contradiction, that the concerning claim does not hold. Thus, we are assuming that there is no positive real constant C = C(W, L) such that (1) is valid. In particular, k = 1 2 N is not such a constant C, so that there exists a function 1,2 n u 2 W (W;R ) such that ku k > 1 (ku k +ke(u )k ). 1 1,2,W 1 0,2,W 1 0,2,W Similarly, k = 2 2 N is not such a constant C, so that there exists a function u 2 1,2 n W (W;R ) such that ku k > 2 (ku k +ke(u )k ). 2 1,2,W 2 0,2,W 2 0,2,W Hence, since no k 2 N is such a constant C, reasoning inductively, for each k 2 N there 1,2 n exists a function u 2 W (W;R ) such that ku k > k(ku k +ke(u )k ). k 1,2,W k 0,2,W k 0,2,W In particular, defining v = ku k k 1,2,W we obtain kv k = 1 > k(kv k +ke(v )k ), k 1,2,W k 0,2,W k 0,2,W so that (kv k +ke(v )k ) < , 8k 2 N. k 0,2,W k 0,2,W From this we obtain kv k < , k 0,2,W and ke (v )k < , 8k 2 N, i j k 0,2,W so that kv k ! 0, as k ! ¥, k 0,2,W and ke (v )k ! 0, as k ! ¥. 0,2,W i j k In particular, k(v ) k ! 0, 8j 2 f1, . . . , ng. k j,j 0,2,W At this point, we recall the following identity in the distributional sense, found in [3], page 12: ¶ (¶ v ) = ¶ e (v) + ¶ e (v) ¶ e (v), 8i, j, l 2 f1, . . . , ng. (2) j l i j il l i j i jl Fix j 2 f1, . . . , ng and observe that k(v ) k  Ck(v ) k , k j 1,2,V k j 1,2,W AppliedMath 2023, 3 409 so that C 1 , 8k 2 N. k(v ) k k(v ) k k j 1,2,V k j 1,2,W Hence, k(v ) k j 1,2,W n o = sup hr(v ) ,rji +h(v ) , ji : kjk  1 2 2 k j k j 1,2,W L (W) L (W) j2C (W) * !+ (v ) k j = r(v ) ,r k j k(v ) k k j 1,2,W L (W) * !+ (v ) k j + (v ) , k(v ) k k j 1,2,W L (W) 0 1 * !+ * !+ (v ) (v ) k j k j @ A C r(v ) ,r + (v ) , k j k j k(v ) k k(v ) k 1,2,V 1,2,V k j k j 2 2 L (V) L (V) n o = C sup hr(v ) ,rji 2 +h(v ) , ji 2 : kjk  1 . (3) 1,2,V k j L (V) k j L (V) j2C (V) Here, we recall that C > 0 is the constant concerning the extension Theorem 1. From such results and (2), we have that n o sup hr(v ) ,rji +h(v ) , ji : kjk  1 2 2 k j k j 1,2,W L (W) L (W) j2C (W) n o C sup hr(v ) ,rji 2 +h(v ) , ji 2 : kjk  1 1,2,V k j L (V) k j L (V) j2C (V) = C sup he (v ), j i +he (v ), j i 2 2 jl k ,l jl k ,l L (V) L (V) j2C (V) he (v ), j i +h(v ) , ji , : kjk  1 . (4) 2 2 ll k ,j k j 1,2,V L (V) L (V) Therefore, k(v ) k k j 1,2 W (W) ( ) = sup fhr(v ) ,rji +h(v ) , ji : kjk  1g 2 2 k j k j 1,2,W L (W) L (W) j2C (W) n o C ke (v )k +ke (v )k +k(v ) k å jl k 0,2,V ll k 0,2,V k j 0,2,V l=1 n o C ke (v )k +ke (v )k +k(v ) k 1 å jl k 0,2,W ll k 0,2,W k j 0,2,W l=1 < , (5) for appropriate C > 0 and C > 0. 1 2 Summarizing, k(v ) k < , 8k 2 N. 1,2 k j (W (W)) From this we obtain kv k ! 0, as k ! ¥, k 1,2,W AppliedMath 2023, 3 410 which contradicts kv k = 1, 8k 2 N. k 1,2,W The proof is complete. n 1 Corollary 1. Let W  R be an open, bounded and connected set with a C boundary ¶W. Define 1,2 n 2 nn e : W (W;R ) ! L (W;R ) by e(u) = fe (u)g i j where e (u) = (u + u ), 8i, j 2 f1, . . . , ng. i j i,j j,i Define also, 1/2 n n ke(u)k = ke k . 0,2,W å å i j(u) 0,2,W i=1 j=1 + n 0 Let L 2 R be such V = [L, L] is also such that W  V . Moreover, define 1,2 n H = fu 2 W (W;R ) : u = 0, on G g, 0 0 where G  ¶W is a measurable set such that the Lebesgue measure m n1(G ) > 0. 0 0 Assume also G is such that for each j 2 f1, , ng and each x = (x , , x ) 2 W there 0 1 n exists x = ((x ) , , (x ) ) 2 G such that 0 0 0 n 0 (x ) = x , 8l 6= j, l 2 f1, , ng, l l and the line A  W x ,x where A = f(x , , (1 t)(x ) + tx , , x ) : t 2 [0, 1]g. x ,x 0 n 1 j j Under such hypotheses, there exists C(W, L) 2 R such that kuk  C(W, L) ke(u)k , 8u 2 H . 1,2,W 0,2,W 0 Proof. Suppose, to obtain contradiction, that the concerning claim does not hold. Hence, for each k 2 N there exists u 2 H such that ku k > k ke(u )k . k 1,2,W k 0,2,W In particular, defining v = ku k k 1,2,W similarly to the proof of the last theorem, we may obtain k(v ) k ! 0, as k ! ¥, 8j 2 f1, . . . , ng. k j,j 0,2,W From this, the hypotheses on G and from the standard Poincaré inequality proof we obtain k(v ) k ! 0, as k ! ¥, 8j 2 f1, . . . , ng. j 0,2,W Thus, also similarly as in the proof of the last theorem, we may infer that kv k ! 0, as k ! ¥, k 1,2,W AppliedMath 2023, 3 411 which contradicts kv k = 1, 8k 2 N. k 1,2,W The proof is complete. 3. An Existence Result for a Non-Linear Model of Plates In the present section, as an application of the results on Korn’s inequalities presented in the previous sections, we develop a new global existence proof for a Kirchhoff–Love thin plate model. Previous results on the existence of mathematical elasticity and related models may be found in [2–4]. At this point we start to describe the primal formulation. Let W  R be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of W, which is assumed to be regular (Lipschitzian), is denoted by ¶W. The vectorial basis related to the cartesian system fx , x , x g is denoted 2 3 by (a , a ), where a = 1, 2 (in general, Greek indices stand for 1 or 2), and where a is the a 3 3 vector normal to W, whereas a and a are orthogonal vectors parallel to W. Furthermore, n is the outward normal to the plate surface. The displacements will be denoted by u ˆ = fu ˆ , u ˆ g = u ˆ a + u ˆ a . a a a 3 3 3 The Kirchhoff–Love relations are u ˆ (x , x , x ) = u (x , x ) x w(x , x ) a a ,a 1 2 3 1 2 3 1 2 and u ˆ (x , x , x ) = w(x , x ). (6) 3 1 2 3 1 2 Here, h/2  x  h/2 so that we have u = (u , w) 2 U where 3 a 1,2 2 2,2 U = (u , w) 2 W (W;R ) W (W), ¶w u = w = = 0 on ¶W ¶n 1,2 2 2,2 = W (W;R ) W (W). 0 0 It is worth emphasizing that the boundary conditions here specified refer to a clamped plate. 2 22 We define the operator L : U ! Y Y, where Y = Y = L (W;R ), by L(u) = fg(u), k(u)g, u + u w w a,b b,a ,a ,b g (u) = + , ab 2 2 k (u) = w . ab ,ab The constitutive relations are given by N (u) = H g (u), (7) ab ablm lm M (u) = h k (u), (8) ab ablm lm where f H g and fh = H g, are symmetric positive definite fourth-order ablm ablm ablm 1 1 tensors. From now on, we denote f H g = f H g and fh g = fh g . ablm ablm ablm ablm AppliedMath 2023, 3 412 Furthermore, fN g denote the membrane force tensor and f M g the moment one. ab ab The plate stored energy, represented by (G L) : U ! R, is expressed by Z Z 1 1 (G L)(u) = N (u)g (u) dx + M (u)k (u) dx (9) ab ab ab ab 2 2 W W and the external work, represented by F : U ! R, is given by F(u) = hw, Pi 2 +hu , P i 2 , (10) a a L (W) L (W) where P, P , P 2 L (W) are external loads in the directions a , a , and a , respectively. The 1 2 3 1 2 potential energy, denoted by J : U ! R, is expressed by J(u) = (G L)(u) F(u) Finally, we also emphasize from now on, as their meaning are clear, we may denote 2 2 22 2 L (W) and L (W;R ) simply by L , and the respective norms bykk . Moreover, deriva- tives are always understood in the distributional sense, 0 may denote the zero vector in appropriate Banach spaces, and the following and relating notations are used: ¶w w = , ,a ¶x ¶ w w = , ,ab ¶x ¶x a b ¶u u = , a,b ¶x ¶N ab N = , ab,1 ¶x and ¶N ab N = . ab,2 ¶x 4. On the Existence of a Global Minimizer At this point, we present an existence result concerning the Kirchhoff–Love plate model. We start with the following two remarks. ¥ 2 Remark 4. Let fP g 2 L (W;R ). We may easily obtain by appropriate Lebesgue integration fT g symmetric and such that ab T = P , in W. ab,b Indeed, extending fP g to zero outside W if necessary, we may set T (x, y) = P (x, y) dx, 11 1 T (x, y) = P (x, x) dx, 22 2 and ˜ ˜ T (x, y) = T (x, y) = 0, in W. 12 21 Thus, we may choose a C > 0 sufficiently big, such that fT g = fT + Cd g ab ab ab AppliedMath 2023, 3 413 is positive definite in W, so that T = T = P , ab,b ab,b a where fd g ab is the Kronecker delta. Therefore, for the kind of boundary conditions of the next theorem, we do not have any restriction for the fP g norm. In summary, the next result is new and it is really a step forward concerning the previous one in Ciarlet [3]. We emphasize that this result and its proof through such a tensor fT g are new, ab even though the final part of the proof is established through a standard procedure in the calculus of variations. Finally, more details on the Sobolev spaces involved may be found in [5–8]. Related duality principles are addressed in [5,7,9]. At this point, we present the main theorem in this section. Theorem 3. Let W  R be an open, bounded, connected set with a Lipschitzian boundary denoted by ¶W = G. Suppose (G L) : U ! R is defined by G(Lu) = G (g(u)) + G (k(u)), 8u 2 U, 1 2 where G (gu) = H g (u)g (u) dx, ablm ab lm and G (ku) = h k (u)k (u) dx, ablm ab lm where L(u) = (g(u), k(u)) = (fg (u)g,fk (u)g), ab ab u + u w w a,b b,a ,a ,b g (u) = + , ab 2 2 k (u) = w , ab ,ab and where J(u) = W(g(u), k(u))hP , u i 2 a a L (W) hw, Pi 2 hP , u i 2 L (W) a L (G ) hP , wi , (11) L (G ) where, 1,2 2 2,2 U = fu = (u , w) = (u , u , w) 2 W (W;R ) W (W) : a 1 2 ¶w u = w = = 0, on G g, (12) ¶n where ¶W = G [ G and the Lebesgue measures 0 t m (G \ G ) = 0, G 0 t and m (G ) > 0. G 0 AppliedMath 2023, 3 414 We also define F (u) = hw, Pi hu , P i hP , u i 2 2 2 1 a a a L (W) L (W) a L (G ) t 2 hP , wi 2 +h# , u i 2 L (G ) a L (G ) t t hu, fi 2 +h# , u i 2 L a L (G ) hu, f i 2 hu , P i 2 +h# , u i 2 , (13) a a a 1 a L L (W) L (G ) where hu, f i = hu, fi hu , P i , 2 2 2 1 a a L L L (W) # > 0, 8a 2 f1, 2g and ¥ 3 f = (P , P) 2 L (W;R ). Let J : U ! R be defined by J(u) = G(Lu) + F (u), 8u 2 U. 22 Assume there exists fc g 2 R such that c > 0, 8a, b 2 f1, 2g and ab ab G (k(u))  c kw k , 8u 2 U. 2 ab ,ab Under such hypotheses, there exists u 2 U such that J(u ) = min J(u). u2U Proof. Observe that we may find T = f(T ) g such that a a b divT = T = P , a ab,b a and also such that fT g is positive, definite, and symmetric (please see Remark 4). ab Thus, defining u + u a,b b,a v (u) = + w w , (14) ,a ab ,b 2 2 we obtain J(u) = G (fv (u)g) + G (k(u))hu, fi 2 +h# , u i 2 2 a 1 ab a L L (G ) = G (fv (u)g) + G (k(u)) +hT , u i 2 hu, f i 2 +h# , u i 2 a a 1 ab 2 ab,b 1 a L (W) L L (G ) u + u a,b b,a = G (fv (u)g) + G (k(u)) T , 1 ab 2 ab L (W) +hT n , u i hu, f i +h# , u i 2 2 2 ab b a 1 a L (G ) L a L (G ) t t = G (fv (u)g) + G (k(u)) T , v (u) w w hu, f i +h# , u i 2 2 1 ab 2 ab ab ,a ,b 1 a L a L (G ) L (W) +hT n , u i a 2 ab b L (G ) 2 2 c kw k + T , w w hu, f i 2 +h# , u i 2 + G (fv (u)g) ab ,ab ab ,a ,b 1 a 1 ab 2 2 L a L (G ) L (W) t hT , v (u)i +hT n , u i . (15) 2 2 ab ab ab b a L (W) L (G ) From this, since fT g is positive definite, clearly J is bounded below. ab Let fu g 2 U be a minimizing sequence for J. Thus, there exists a 2 R such that lim J(u ) = inf J(u) = a . n!¥ u2U AppliedMath 2023, 3 415 From (15), there exists K > 0 such that k(w ) k < K ,8a, b 2 f1, 2g, n 2 N. n ,ab 2 1 2,2 Therefore, there exists w 2 W (W) such that, up to a subsequence not relabeled, (w ) * (w ) , weakly in L , ,ab 0 ,ab 8a, b 2 f1, 2g, as n ! ¥. Moreover, also up to a subsequence not relabeled, 2 4 (w ) ! (w ) , strongly in L and L , (16) n ,a 0 ,a 8a,2 f1, 2g, as n ! ¥. Furthermore, from (15), there exists K > 0 such that, k(v ) (u)k < K ,8a, b 2 f1, 2g, n 2 N, ab 2 2 and thus, from this, (14) and (16), we may infer that there exists K > 0 such that k(u ) + (u ) k < K ,8a, b 2 f1, 2g, n 2 N. n n a,b b,a 2 3 From this and Korn’s inequality, there exists K > 0 such that ku k  K , 8n 2 N. 1,2 2 n 4 W (W;R ) 1,2 2 Therefore, up to a subsequence not relabeled, there exists f(u ) g 2 W (W,R ), such that 0 a (u ) + (u ) * (u ) + (u ) , weakly in L , n a,b n b,a 0 a,b 0 b,a 8a, b 2 f1, 2g, as n ! ¥, and (u ) ! (u ) , strongly in L , n a 0 a 8a 2 f1, 2g, as n ! ¥. Moreover, the boundary conditions satisfied by the subsequences are also satisfied for w and u in a trace sense, so that 0 0 u = ((u ) , w ) 2 U. 0 0 a 0 From this, up to a subsequence not relabeled, we obtain g (u ) * g (u ), weakly in L , ab n ab 0 8a, b 2 f1, 2g, and k (u ) * k (u ), weakly in L , ab ab 0 8a, b 2 f1, 2g. Therefore, from the convexity of G in g and G in k, we obtain 1 2 inf J(u) = a u2U = lim inf J(u ) n!¥ J(u ). (17) Thus, J(u ) = min J(u). u2U AppliedMath 2023, 3 416 The proof is complete. 5. Conclusions In this article, we have developed a new proof for Korn’s inequality in a specific n-dimensional context. In the second text part, we present a global existence result for a non-linear model of plates. Both results represent some new advances concerning the present literature. In particular, the results for Korn’s inequality known so far are for a three-dimensional context such as in [1], for example, whereas we have here addressed a more general n-dimensional case. In a future research, we intend to address more general models, including the corre- sponding results for manifolds in R . Funding: This research received no external funding Conflicts of Interest: The author declares no conflict of interest. References 1. Lebedev, L.P.; Cloud, M.J. Korn’s Inequality. In Encyclopedia of Continuum Mechanics; Altenbach, H., Öchsner, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2020. [CrossRef] 2. Ciarlet, P. Mathematical Elasticity; Vol. I—Three Dimensional Elasticity; Elsevier: Amsterdam, The Netherlands, 1988. 3. Ciarlet, P. Mathematical Elasticity; Vol. II—Theory of Plates; Elsevier: Amsterdam, The Netherlands, 1997. 4. Ciarlet, P. Mathematical Elasticity; Vol. III—Theory of Shells; Elsevier: Amsterdam, The Netherlands, 2000. 5. Botelho, F.S. Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering; CRC Taylor and Francis: Uttar Pradesh, India, 2020. 6. Adams, R.A.; Fournier, J.F. Sobolev Spaces, 2nd ed.; Elsevier: New York, NY, USA, 2003. 7. Botelho, F.S. Functional Analysis and Applied Optimization in Banach Spaces; Springer: Cham, Switzerland, 2014. 8. Evans, L.C. Partial Differential Equations. In Graduate Studies in Mathematics; AMS: Providence, RI, USA, 1998. 9. Ekeland, I.; Temam, R. Convex Analysis and Variational Problems; Elsevier: Amsterdam, The Netherlands, 1976. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Journal

AppliedMath – Multidisciplinary Digital Publishing Institute

Published: May 2, 2023

Keywords: Korn’s inequality; global existence result; non-linear plate model

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