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L. Tentarelli, P. Tilli (2017)
De Giorgi’s approach to hyperbolic Cauchy problems: The case of nonhomogeneous equationsCommunications in Partial Differential Equations, 43
Università degli Studi di Genova, Via all'Opera Pia, 15 -16145 Genova Italy
E. De Giorgi (1996)
Conjectures concerning some evolution problems. A celebration of John F. Nash, jrDuke Math. J., 81
U. Stefanelli (2010)
THE DE GIORGI CONJECTURE ON ELLIPTIC REGULARIZATIONMathematical Models and Methods in Applied Sciences, 21
M. Liero, U. Stefanelli (2013)
A New Minimum Principle for Lagrangian MechanicsJournal of Nonlinear Science, 23
(2013)
Weighted inertia-dissipation-energy functionals for semilinear equations
E. Serra, P. Tilli (2013)
A minimization approach to hyperbolic Cauchy problemsarXiv: Analysis of PDEs
W. Lai, D. Rubin, E. Krempl (2009)
Introduction to Continuum Mechanics
J. Nash, H. Kuhn, L. Nirenberg, P. Sarnak, Morris Weisfeld (1996)
A celebration of John F. Nash, Jr.
L. Tentarelli, P. Tilli (2018)
An existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approachJournal of Differential Equations
E. Serra, P. Tilli (2012)
Nonlinear wave equations as limits of convex minimization problems: proof of a conjecture by De GiorgiAnnals of Mathematics, 175
E. De Giorgi (1996)
10.1215/S0012-7094-96-08114-4Duke Math. J., 81
Dipartimento di Ingegneria We show that the solution of Cauchy problem for the classical ODE my = f can be Meccanica, Energetica, Gestionale e dei Trasporti (DIME), Università degli obtained as the limit of minimizers of exponentially weighted convex variational Studi di Genova, Via all’Opera Pia, integrals. This complements the known results about weighted inertia-energy 15, 16145, Genova, Italy approach to Lagrangian mechanics and hyperbolic equations. MSC: 49J45; 70H30 Keywords: Calculus of variations; Newton’s second law; Weighted variational integrals 1 Introduction and statement of the result ∞ + N N N Let f ∈ L (R ; R ), u ∈ R , v ∈ R ,and m > 0. Let us consider the Cauchy problem 0 0 my = f, t >0, (1.1) y(0) = u , y (0) = v , 0 0 governing the motion of a material point of mass m subject to the force field f.Our goal is to show that the solution to (1.1)isthe limitas h → +∞ of the minimizers of the following + N functionals defined on trajectories y : R → R : +∞ +∞ –ht –ht y (t) e dt – f (t) · y(t)e dt, h ∈ N, 2h 0 0 ∞ + N subject to the same initial conditions, where (f ) ⊂ L (R ; R )is a sequence such that h h∈N ∗ ∞ + N f f in w – L (R ; R )as h → +∞. More precisely, letting +∞ 2,1 + N –t A := v ∈ W R ; R : v (t) e dt <+∞ loc © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Percivale and Mainini Advances in Continuous and Discrete Models (2023) 2023:20 Page 2 of 9 ∞ + N and f ∈ L (R ; R )for every h ∈ N, we may define the rescaled energy functional (see also Lemma 2.3) +∞ +∞ 2 –t –2 –1 –t |u (t)| e dt – h f (h t) · u(t)e dt ⎪ h 2 0 0 J (u):= if u ∈ A, 2,1 + N +∞ otherwise in W (R ; R ). loc We will prove the following result. Theorem 1.1 For every h ∈ N, there exists a unique solution u to the problem –1 min J (u): u ∈ A, u(0) = u , u (0) = h v . h 0 0 ∗ ∞ + N Moreover, if f f in w – L (R ; R ) as h → +∞, then by setting y (t):= u (ht) we have h h ∗ 2,∞ N + y y in w – W ((0, T); R ) for every T >0, where y is the unique solution on R of problem (1.1). A variational approach based on the minimization of weighted inertia-energy (WIE) functionals can be used for approximating large classes of initial value problems of the second order. An example is the nonhomogeneous wave equation + N w = w + g in R × R . tt 2 2 N 1 N Indeed, it has been shown in [8]thatgiven g ∈ L ((0, +∞); L (R )), α ∈ H (R ), and loc 1 N 2 2 N β ∈ H (R ), there exists a sequence (g ) converging to g in L ((0, T); L (R )) for every h h∈N T > 0 such that the following properties hold. First, the WIE functional +∞ 1 1 2 2 –t –2 –2 –1 e u (t, x) + h ∇u(t, x) – h g h t, x u(t, x) dt dx tt h N 2 2 0 R 1 + N has, for every h ∈ N, a unique minimizer u in the class of functions u ∈ L (R × R ) loc such that 1 + N 1 + N ∇u ∈ L (R × R ), u ∈ L (R × R ), loc loc +∞ –t 2 2 e {|u | + |∇u| } dt dx <+∞, N tt 0 R –1 u(0, x)= α(x), u (0, x)= h β(x). Second, by setting w (t, x):= u (ht, x)the sequence (w ) converges weakly in h h h h∈N 1 N H ((0, T) × R )for every T >0 to a function w that solves in the sense of distributions in + N R × R the initial value problem w = w + g, tt (1.2) w(0, x)= α(x), w (0, x)= β(x). A similar result holds for other classes of hyperbolic equations as shown in [4, 6, 9]. In p–2 particular, it applies to the nonlinear wave equation w = w – |w| w, p ≥ 2, as conjec- tt tured by De Giorgi [1]and first proven in [5]; see also [7]. Let us mention that (the scalar Percivale and Mainini Advances in Continuous and Discrete Models (2023) 2023:20 Page 3 of 9 version of) Theorem 1.1 is not a direct consequence of the above result from [8], since we should apply the latter to constant-in-space forcing terms g and initial data α, β,and since the approximating sequence (g ) in [8] is not arbitrary but obtained by means of h h∈N a specific construction, not allowing, for instance, for the choice g ≡ g for every h. Concerning the WIE approach for ODEs, let us mention its application in [3] for pro- viding a variational approach to Lagrangian mechanics by considering an equation of the form my + ∇U(y)=0, t > 0, (1.3) 1 N for given potential energy U ∈ C (R ), bounded from below and m >0. The main theo- rem of [3] proves indeed that solutions to the initial value problem for (1.3) can be approx- imated by rescaled minimizers, subject to the same initial conditions, of the functionals +∞ –t –2 G (v)= e v (t) + h U v(t) dt, h ∈ N. It is worth noticing that also in this case, Theorem 1.1 is not a consequence of the result from [3] since the latter requires that the force field is conservative and independent of t. We have already observed that in the scalar case, problem (1.1) is a particular case of problem (1.2), obtained by taking constant initial data and letting the forcing term de- pend only on time. Let us also mention another interpretation of (1.1) from a continuum mechanics point of view. Indeed, Newton’s second law (1.1) governs the motion of the center of mass of a body occupying a reference configuration ⊂ R . In more detail, let ρ be the mass density of the body, and let u(t, x) be the position of the material point x at time t.If T is the Cauchy stress tensor and b is the body force field acting on , then the equation of motion (see,e.g., [2]) takes the form ρu = div T + b in R × . (1.4) tt Therefore by integrating in both sides of (1.4) we formally get N–1 ρu dx = div T dx + bdx = T · n dH + b dx =: f , t >0, dt that is, m y = f , t >0, where f = f (t) is the total force acting on the body, accounting for surface and body forces, m = ρ(x) dx is the mass of the body, and –1 y(t)= m ρ(x)u(t, x) dx is the position at time t of the center of mass of the body during the motion. Therefore Newton’s second law (1.1) can be viewed as the average in space of the equation of motion Percivale and Mainini Advances in Continuous and Discrete Models (2023) 2023:20 Page 4 of 9 (1.4). In this perspective, Theorem 1.1 can be seen as a result about the equation of motion in R in the above average sense. Let us finally stress that the methods described in this paper, here only devoted to the elementary problem (1.1), can be extended to nonlinear problems like y = ∇ G(t, y)under suitable assumptions on G and also to hyperbolic problems such as (1.2) allowing us to get further results on these topics. In this perspective, we will develop our analysis in a forthcoming paper. 2 Existence of minimizers In this section, we provide some preliminary results we are going to use for proving The- 2,2 N orem 1.1. First of all, it is worth noticing that if u ∈ A,then u ∈ W ((0, T); R )for every T >0, hence both u(0) and u (0) are well defined. Moreover, if u ∈ A, then by the Cauchy– Schwarz inequality, 1/2 +∞ +∞ –1 –t –t f h t · u(t)e dt ≤ f u(t) e dt , 0 0 and the integral in the left-hand side is finite (see Lemma 2.1), so that J (u) is well-defined and finite. In fact, we have the following estimates. –t/2 2 N –t/2 2 N Lemma 2.1 Let u ∈ A. Then e u ∈ L ((0, +∞); R ), e u ∈ L ((0, +∞); R ), and +∞ +∞ 2 2 2 –t –t u (t) e dt ≤ 2 u (0) +4 u (t) e dt, (2.1) 0 0 +∞ +∞ 2 2 2 2 –t –t u(t) e dt ≤ 2 u(0) +8 u (0) +16 u (t) e dt. (2.2) 0 0 N N Proof We have u ∈ AC([0, T]; R )and u ∈ AC([0, T]; R )for every T > 0. Therefore d 2 d 2 |u(t)| =2u(t) · u (t)and |u (t)| =2u (t) · u (t)for a.e. t > 0. Moreover, given T >0, dt dt we integrate by parts and obtain T T 2 2 T –t –t –t/2 –t/2 u (t) e dt = –e u (t) +2 e u (t) · u (t)e dt 0 0 T T 2 2 2 –t –t ≤ u (0) + u (t) e dt +2 u (t) e dt, 0 0 wherewehaveusedthe Younginequality. By letting T → +∞ we get (2.1). The same computation entails T T T 2 2 2 2 –t –t –t u(t) e dt ≤ u(0) + u(t) e dt +2 u (t) e dt. 0 0 0 By letting T → +∞ and by taking advantage of (2.1)weobtain(2.2). The next lemma proves the first statement of Theorem 1.1. Lemma 2.2 For every h ∈ N, there exists a unique solution to the problem –1 min J (u): u ∈ A, u(0) = u , u (0) = h v . (2.3) h 0 0 Percivale and Mainini Advances in Continuous and Discrete Models (2023) 2023:20 Page 5 of 9 Proof We first observe that J is strictly convex and that the minimization set is convex. Therefore if a minimizer exists, then it is necessarily unique, so we are left to prove the –1 existence. If u ∈ A is such that u(0) = u and u (0) = h v , then Lemma 2.1 entails 0 0 +∞ +∞ 2 2 –t 2 –2 2 –t u(t) e dt ≤ 2|u | +8h |v | +16 u (t) e dt (2.4) 0 0 0 0 and +∞ +∞ 2 2 –t –2 2 –t u (t) e dt ≤ 2h |v | +4 u (t) e dt. (2.5) 0 0 –1 Let (u ) be a minimizing sequence for problem (2.3). Since u + h tv is admissible for k k∈N 0 0 problem (2.3), we have, for any k large enough, –1 J (u ) ≤ J u + h tv +1, h k h 0 0 whence by (2.4) and by the Young and Cauchy–Schwarz inequalities, denoting by C vari- ous constants only depending on f , h, u , v , m,weget h ∞ 0 0 +∞ –t u (t) e dt +∞ –2 –1 –t ≤ h f h t · u (t)e dt h k +∞ 2 2 –2 –1 –1 –t – h f h t · u + h tv e dt + h 0 0 m m +∞ –2 –t ≤ f h u (t) e dt + C h ∞ k (2.6) +∞ –2 –t ≤ f h u (t) e dt + C h ∞ k +∞ 1 32 –t –4 2 ≤ u (t) e dt + h f + C k h 32 m +∞ –t ≤ u (t) e dt + C. By taking into account of (2.4), (2.5), and (2.6) we get that the sequence (e u ) is k k∈N 2,2 + N 2,2 + N equibounded in W (R ; R ). So there exists v ∈ W (R ; R ) such that, up to extracting t t – 2,2 + N 2,2 N 2 2 asubsequence, e u v in W (R ; R ), and hence u u := e v in W ((0, T); R ) k k –1 for every T >0, and u(0) = u , u (0) = h v . Therefore for every T >0, we have 0 0 +∞ T T 2 2 2 –t –t –t lim inf u (t) e dt ≥ lim inf u (t) e dt ≥ u (t) e dt, k k k→+∞ k→+∞ 0 0 0 and hence +∞ T +∞ 2 2 2 –t –t –t u (t) e dt = sup u (t) e dt ≤ lim inf u (t) e dt, k→+∞ 0 T>0 0 0 Percivale and Mainini Advances in Continuous and Discrete Models (2023) 2023:20 Page 6 of 9 so eventually we find u ∈ A, and since +∞ +∞ –2 –1 –t –2 –1 –t/2 lim h f h t · u e dt = h f h t · ve dt h k h k→+∞ 0 0 +∞ –2 –1 –t = h f h t · ue dt, we get lim inf J (u ) ≥ J (u). h k h k→+∞ We conclude that u is a solution to (2.3). Lemma 2.3 Let h ∈ N. If u is the unique solution to (2.3), then y (t):= u (ht) is the unique h h minimizer of +∞ +∞ ⎨ m 2 –ht –ht |y (t)| e dt – f (t) · y(t)e dt if y ∈ A , h h 2h 0 0 F (y):= 2,1 ⎩ + N +∞ otherwise in W (R ; R ) loc over A , where +∞ 2,1 + N –ht A := y ∈ W R ; R : y (t) e dt <+∞, y(0) = u , y (0) = v . h 0 0 loc –1 Proof Since u ∈ A and u (0) = u , u (0) = h v , we directly see that y ∈ A and h h 0 0 h h h –1 –1 h F (y )= J (u ). Moreover, if y ∈ A , then by setting u (t)= y(h t)we get u ∈ h h h h h h –1 –1 A, u (0) = u , u (0) = h v ,and h F (y)= J (u ). Therefore F (y ) ≤ F (y)for every h 0 0 h h h h h h h y ∈ A , and equality holds if and only if y = y ,asclaimed. 3 Proof of Theorem 1.1 Given y minimizing F over A , here we prove suitable boundedness estimates for the h h sequence (y ) , which are the main step toward the proof of Theorem 1.1. h∈N ∞ + N Lemma 3.1 For every h ∈ N, let y be as in Lemma 2.3. Then y ∈ L (R ; R ), and h h –1 y ≤ m sup f . h ∞ h∈N 2,∞ N Moreover, the sequence (y ) is equibounded in W ((0, T); R ) for every T >0. h∈N + N Proof Let h ∈ N, ϕ ∈ C (R ; R ), and let ξ be the unique solution to ⎨ t ξ = e ϕ, t >0, ξ(0) = ξ (0) = 0. –2 By setting ψ (t):= h ξ(ht)we see that ψ (0) = ψ (0) = 0 and h h h +∞ +∞ 2 2 –ht –1 t ψ (t) e dt = h ϕ(t) e dt, 0 0 Percivale and Mainini Advances in Continuous and Discrete Models (2023) 2023:20 Page 7 of 9 + N and the integral in the right-hand side is finite since ϕ ∈ C (R ; R ). Thus we get y + ψ ∈ h h A . The minimality of y entails the validity of the first-order relation +∞ +∞ –2 –ht –ht mh y (t) · ψ (t)e dt = f (t) · ψ (t)e dt. (3.1) h h h 0 0 Since ξ(0) = 0, using integration by parts, we have, for every ν >0 and every τ >0, τ τ –t –t ξ(t) e dt ≤ ξ(t) + ν e dt 0 0 ξ (t) · ξ(t) 2 τ –t –t = –e ξ(t) + ν + e dt 2 2 0 |ξ(t)| + ν –t ≤ ν + ξ (t) e dt, Then by the arbitrariness of ν and τ , repeating the same argument taking into account that ξ (0) = 0, we obtain +∞ +∞ +∞ –t –t –t ξ(t) e dt ≤ ξ (t) e dt ≤ ξ (t) e dt. 0 0 0 Therefore +∞ –ht f (t) · ψ (t)e dt +∞ –3 –1 –s = h f h s · ξ(s)e ds (3.2) +∞ +∞ –3 –s –3 ≤ h f e ξ (s) ds = h f ϕ(s) ds. h ∞ h ∞ 0 0 We recall from Lemma 2.3 that y (t)= u (ht), where u is the unique solution to (2.3). h h Hence, taking into account that +∞ –2 –ht h y (t) · ψ (t)e dt +∞ –ht = u (ht) · ξ (ht)e dt +∞ +∞ –1 –s –1 = h u (s) · ξ (s)e ds = h u (s) · ϕ(s) ds h h 0 0 and using (3.1)and (3.2), we get +∞ +∞ –1 –2 u (s) · ϕ(s) ds ≤ m h f |ϕ| ds. (3.3) h ∞ 0 0 + N + N 1 + N Since ϕ ∈ C (R ; R ) is arbitrary and C (R ; R )is dense in L (R ; R ), (3.3)entails c c u ≤ f , h ∞ ∞ 2 h m Percivale and Mainini Advances in Continuous and Discrete Models (2023) 2023:20 Page 8 of 9 that is, y ≤ f . (3.4) h ∞ Eventually, we have, for every t ∈ [0, T], t t y (t)= v + y (s) ds and y (t)= u + tv + (t – s)y (s) ds, 0 0 0 h h h h 0 0 and hence (3.4)yields y ≤|u | + T |v | + f (3.5) L (0,T) 0 0 h ∞ 2m and y ≤|v | + f . (3.6) 0 h ∞ h ∞ L (0,T) Estimates (3.4), (3.5), and (3.6)prove theresult, sincethe sequence (f ) is bounded h h∈N ∞ + in L (R ). Proof of Theorem 1.1 For every h ∈ N,let y be as in Lemma 2.3.Let T >0, and let ξ ∈ ∞ ht C (R)with spt ξ ⊂ (0, T). Then setting ϕ (t):= ξ(t)e and taking into account the first- order minimality condition (3.1), we have –2 –1 –m y (t) · h ξ (t)+2h ξ (t)+ ξ (t) dt –2 –ht =–mh y (t) · ϕ (t)e dt h h (3.7) T +∞ –2 –ht –2 –ht = mh y (t) · ϕ (t)e dt = mh y (t) · ϕ (t)e dt h h h h 0 0 +∞ T –ht = f (t) · ϕ (t)e dt = f (t) · ξ(t) dt. h h h 0 0 2,∞ N By Lemma 3.1 there exists y ∈ W ((0, T); R ) such that, up to subsequences, y x in ∗ 2,∞ N w – W ((0, T); R ). Therefore we get x(0) = u and x (0) = v . Taking into account (3.7) 0 0 ∗ ∞ + and the w – L (R ) convergence of f to f,inthe limitas h → +∞,weobtain T T –m y (t) · ξ (t) dt = f(t) · ξ(t) dt. 0 0 Thelatterholds forevery ξ ∈ C (R)with spt ξ ⊂ (0, T), and therefore x is the unique solution of my = f, y(0) = u , y (0) = v , 0 0 Percivale and Mainini Advances in Continuous and Discrete Models (2023) 2023:20 Page 9 of 9 ∗ 2,∞ N on [0, T]. Hence the whole sequence (y ) is such that y x in w – W ((0, T); R ). h∈N h h Since the Cauchy problem (1.1)has auniquesolution y on R and since T is arbitrary, we ∗ 2,∞ N conclude that y y in w – W ((0, T); R )as h → +∞ for every T >0, thus proving the theorem. 4Conclusions The paper contains some new ideas concerning the approximation of the solution of the equation of motion of a body with the minimizers of a sequence of variational problems even in the presence of environmental forces. The method can be extended to hyperbolic equations. Acknowledgements Not applicable. Funding EM and DP are supported by the MIUR-PRIN project 2017TEXA3H. EM is supported by the Istituto Nazionale di Alta Matematica project CUP_E55F22000270001. Availability of data and materials Not applicable. Declarations Competing interests The authors declare that they have no competing interests. Author contributions All authors read and approved the final manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 20 December 2022 Accepted: 23 March 2023 References 1. De Giorgi, E.: Conjectures concerning some evolution problems. A celebration of John F. Nash, jr. Duke Math. J. 81, 61–100 (1996) 2. Gurtin, M.: An Introduction to Continuum Mechanics. Springer, Berlin (1999) 3. Liero, M., Stefanelli, U.: A new minimum principle for Lagrangian mechanics. J. Nonlinear Sci. 23(2), 179–204 (2013) 4. Liero, M., Stefanelli, U.: Weighted inertia-dissipation-energy functionals for semilinear equations. Boll. Unione Mat. Ital. 9(6), 1–27 (2013) 5. Serra, E., Tilli, P.: Nonlinear wave equation as limits of convex minimization problems: proof of a conjecture by De Giorgi. Ann. Math. 175, 1551–1574 (2012) 6. Serra, E., Tilli, P.: A minimization approach to hyperbolic Cauchy problems. J. Eur. Math. Soc. 18(9), 2019–2044 (2016) 7. Stefanelli, U.: The De Giorgi conjecture on elliptic regularization. Math. Models Methods Appl. Sci. 21, 1377–1394 (2011) 8. Tentarelli, L., Tilli, P.: De Giorgi’s approach to hyperbolic Cauchy problems: the case of nonhomogeneous equations. Commun. Partial Differ. Equ. 43(4), 677–698 (2018) 9. Tentarelli, L., Tilli, P.: An existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach. J. Differ. Equ. 266(4), 5185–5208 (2019)
Advances in Continuous and Discrete Models – Springer Journals
Published: Apr 28, 2023
Keywords: Calculus of variations; Newton’s second law; Weighted variational integrals; 49J45; 70H30
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