In this talk we present an overview on the extensions of the De Giorgi approach to general second order nonlinear hyperbolic equations. We start with an introduction to the original conjecture by E. De Giorgi ([1, 2]) and to its solution by E. Serra and P. Tilli ([4]). Then, we discuss a first extension of this idea (Serra&Tilli, [5]) aimed at investigating a wide class of homogeneous equations. Finally, we announce a further extension to nonhomogeneous equations, obtained by the author in [9] in collaboration with P. Tilli

Tentarelli, L.

In this talk we present an overview on the extensions of the De Giorgi approach to general second order nonlinear hyperbolic equations. We start with an introduction to the original conjecture by E. De Giorgi ([1, 2]) and to its solution by E. Serra and P. Tilli ([4]). Then, we discuss a first extension of this idea (Serra&amp;Tilli, [5]) aimed at investigating a wide class of homogeneous equations. Finally, we announce a further extension to nonhomogeneous equations, obtained by the author in [9] in collaboration with P. Tilli

Tentarelli L.

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEOn the extensions of the De Giorgi approach to nonlinear hyperbolic equations / Tentarelli, L.. - In: RENDICONTI DELSEMINARIO MATEMATICO. - ISSN 0373-1243. - 74:3-4(2016), pp. 151-160.OriginalOn the extensions of the De Giorgi approach to nonlinear hyperbolic equationsPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2771915 since: 2020-05-21T18:20:31ZRendiconti del Seminario MatematicoRend. Sem. Mat. Univ. Politec. TorinoBruxelles-Torino Talks in PDE’s –Turin, May 2–5, 2016Vol. 74, 2 (2016), 151 – 160L. Tentarelli*ON THE EXTENSIONS OF THE DE GIORGI APPROACHTO NONLINEAR HYPERBOLIC EQUATIONSAbstract. In this talk we present an overview on the extensions of the De Giorgi approachto general second order nonlinear hyperbolic equations. We start with an introduction to theoriginal conjecture by E. De Giorgi ([1, 2]) and to its solution by E. Serra and P. Tilli ([4]).Then, we discuss a first extension of this idea (Serra&Tilli, [5]) aimed at investigating a wideclass of homogeneous equations. Finally, we announce a further extension to nonhomoge-neous equations, obtained by the author in [9] in collaboration with P. Tilli.1. De Giorgi’s conjecture.In 1996, E. De Giorgi stated the following conjecture on weak solutions of the defo-cusing NLW equation.CONJECTURE 1 (De Giorgi, [1, 2]). Let w0,w1 C0 Rn , let k 1 be an inte-ger; for every positive real number ε, letwε wε t,x be the minimizer of the functional(1.1) Fε u :0 Rne t ε u t,x 2 1ε2 ∇u t,x2 1ε2 u t,x2k dxdtin the class of all u satisfying the initial conditions(1.2) u 0,x w0 x , u 0,x w1 x .Then, there exists limε 0wε t,x w t,x , satisfying the equation(1.3) w ∆w kw2k 1.REMARK 1. In the statement of the conjecture, we maintained the original for-mulation of [1, 2] and we only changed notation, according to that we use in the sequel.The same thing holds for all the results we mention in this paper. In addition, we recallthat u t,x denotes ut t,x and that, for the sake of simplicity, we always omit thedependence of the functional spaces on Rn, i.e. H1 H1 Rn , Lp Lp Rn and so on.In order to better understand the meaning of the conjecture, it is worth stressingsome characteristic features of the functional Fε.First we note that it involves second order time derivatives. Thus, a minimizerof Fε solves a fourth order PDE. However, if one computes the formal Euler–Lagrange*Author supported by the FIR grant 2013 “Condensed Matter in Mathematical Physics (Cond-Math)”(code RBFR 13WAET).151152 L. Tentarelliequation satisfied by a minimizer wε, then one obtainsε2 e t εwε e t ε ∆wε kw2k 1εand thus, expanding and dropping e t ε,(1.4) ε2wε 2εwε wε ∆wε kw2k 1ε .Consequently, if one assumes that wε w in some suitable sense and lets ε 0, thenone formally obtains (1.3).On the other hand, we also remark that, as Fε is defined through integrals overthe “space–time” 0, Rn, the initial conditions of the Cauchy problem are in factboundary conditions for the minimization problem.In addition, it is convenient to stress the singular nature of the integration weighte t ε. More precisely, one can see that ε 1 e t ε dt is an approximate Dirac delta mea-sure and hence, at least formally,εFε uRn∇w0 x 2 w0 x 2k dx, as ε 0.Hence, this prevents a straightforward application of classical techniques of variationalconvergence, such as Γ–convergence. The previous asymptotic expansion, indeed,shows that this technique does not provide useful information on the limit behaviorof the sequence of the minimizers.Finally, one can note that Fε is convex (for fixed ε 0) and that therefore, upto some suitable technical adaptation, the proof of the existence and uniqueness of theminimizers is not a demanding issue.We also recall that the existence of global solutions for the Cauchy problem(1.3)&(1.2) is not new (see e.g. [7] and the references therein). Actually, as highlightedin [3, 4], the originality of the strategy hinted by De Giorgi lies in how he intended toexploit techniques from the Calculus of Variations. The variational approaches to thewave equation w ∆w and its nonlinear variants that can one can find in the literature(see e.g. [7, 8] and references therein) are based on the interpretation of w ∆w asthe Euler–Lagrange equation of the functionalI w :0 Rnw t,x 2 ∇w t,x 2 dxdt(with possibly lower order terms like w 2k). However, since I is neither convex norbounded from below, one is forced to search for critical points rather than global mini-mizers. Unfortunately, functionals like I behave badly also for the application of Criti-cal Point Theory, so that only partial results can be proved. De Giorgi, on the contrary,introduces a new functional Fε that is quite easy to minimize (regardless of the magni-tude of k) and thus moves the problem to the investigation of the limit behavior of thesequence of the minimizers.On the extensions of the De Giorgi approach 1532. The proof of the conjecture.In 2012, E. Serra and P. Tilli showed that Conjecture 1 is in fact true. Precisely, in [4],they proved the following theorem.THEOREM 1 (Serra&Tilli, [4]). For p 2 and ε 0, let wε t,x denote theunique minimizer of the strictly convex functionalFε u0 Rne t ε u t,x 2 1ε2 ∇u t,x2 1ε2 u t,xp dxdtunder the boundary conditions (1.2), where w0 and w1 are given functions such thatw0, w1 H1 Lp.Then:(a) Estimates. There exists a constant C (which depends only on w0, w1, p and n)such that, for every ε 0,1 ,T0 Rn∇wε t,x 2 wε t,x p dxdt CT, T ε,Rnwε t,x2 dx C andRnwε t,x 2 dx C 1 t2 , t 0,and, for every function h H1 LpRnwε t,x h x dx C h Lp ∇h L2 , for a.e. t 0.(b) Convergence. Every sequence wεi (with εi 0) admits a subsequence which isconvergent, in the strong topology of Lq 0,T A for every T 0 and everybounded open set A Rn (with arbitrary q 2, p if p 2 and q p if p 2),almost everywhere in R Rn and in the weak topology of H1 0,T Rn forevery T 0, to a function w such thatw L R ;Lp , ∇w L R ;L2 ,w L R ;L2 , w L 0,T ;H1 T 0,which solves in R Rn the nonlinear wave equation(2.1) w ∆w p2 wp 2wwith initial conditions as in (1.2).154 L. Tentarelli(c) Energy inequality. LettingE t :Rnw t,x 2 ∇w t,x 2 w t,x p dx,the function w t,x satisfies the energy inequalityE t E 0Rnw1 x 2 ∇w0 x 2 w0 x p dx, for a.e. t 0.REMARK 2. We stress the fact that, in (b), the limit function w solves (2.1) in adistributional (or weak) sense, namely0 Rnw t,x ϕ t,x dxdt0 Rn∇w t,x ∇ϕ t,x dxdt0 Rnp2 w t,xp 2w t,x ϕ t,x dxdtfor every ϕ C0 R Rn . In the sequel we only deal with this type of solutions.Some comments are in order. First, CONJECTURE 1 deals with the nonlinearityw 2k with k integer, while THEOREM 1 treats w p without the assumption of p integer.Another relevant feature of THEOREM 1, is that the assumptions on the initial dataw0, w1 are much weaker than those of the conjecture.On the other hand, the convergence of the sequence of the minimizers is ob-tained up to extracting subsequences, thus “losing” the uniqueness claimed in the con-jecture. In particular, it is an open problem to avoid the extraction of subsequenceswhen p is large.In addition, THEOREM 1 establishes an estimate for the mechanical energy Eusually associated with (2.1), which proves that the obtained solutions are of energyclass in the sense of Struwe (see [8]). When p is “sufficiently” small, the inequality isin fact an equality, whereas, when p is large, energy conservation is still open.For the sake of completeness, we mention that [6] discusses a simplified versionof the conjecture on bounded intervals. However, that paper only deals with the proofof (2.1) and does not treat the fulfillment of the initial condition w 0,x w1 x .3. Extension to homogeneous equations.Now, one can easily see that, settingW vRn12∇v 2 1pv p dx,up to some multiplicative constants equation (2.1) reads(3.1) w t,x ∇W w t, x ,On the extensions of the De Giorgi approach 155where∇W denotes theGâteaux derivative of the functionalW . Therefore, it is naturalto wonder if the sequence of the minimizers of the functional Fε, that here is defined by(3.2) Fε u :t0e t εRnε2 u t,x2dx W u t, dt,converges to a solution of the Cauchy problem associated with (3.1), even for differentchoices ofW .REMARK 3. In (3.2) one uses a different scaling in ε, with respect to (1.1). Thisis due to the fact that in the abstract framework this choice simplifies computations.However, this does not yield significant differences.This problem has been solved again by E. Serra and P. Tilli, in [5]. Beforeshowing the statements of the main results, it is necessary to point out under whichassumptions on the functionalW (that we refer to as assumption (H) in the following),they are valid.(H) The functionalW : L2 0, is lower semi–continuous in the weak topologyof L2, i.eW v liminfkW vk , whenever vk v in L2.Moreover, we assume that the set of functionsW v L2 :W vis a Banach space such thatC0 W L2 (dense embeddings).Finally, W is Gâteaux differentiable on W and its derivative ∇W : W Wsatisfies∇W v W C 1 W v θ , v W,for suitable constants C 0 and θ 0,1 .REMARK 4. Assumption (H) is typically satisfied by standard functionals likeW v 1p Rn∇kv p dx, p 1,(with possibly lower order terms) where W is the space of the L2 functions v with∇kv Lp.THEOREM 2 (Serra&Tilli, [5]). Given w0, w1 W and ε 0,1 , under assump-tion (H) the functional Fε defined in (3.2) has a minimizer wε in the space H2loc 0, ;L2subject to (1.2). Moreover:156 L. Tentarelli(a) Estimates. There exists a constant C, independent of ε, such thatτ TτW wε t, dt CT, τ 0, T ε,Rnwε t,x2 dx C andRnwε t,x 2 dx C 1 t2 , t 0,wε L R ;W C.(b) Convergence. Every sequence wεi (with εi 0) admits a subsequence which isconvergent, in the weak topology of H1 0,T ;L2 for every T 0, to a functionw such thatw H1loc 0, ;L2 , w L R ;L2 , w L R ;W .Moreover, w satisfies the initial conditions (1.2).(c) Energy inequality. Letting(3.3) E t : 12 Rnw t,x 2 dx W w t, ,the function w t,x satisfies the energy inequalityE t E 0 12 Rnw1 x 2 dx W w0 for a.e. t 0.Unfortunately, under these assumptions, it is not known whether w satisfies(3.1). Anyway, Serra&Tilli, still in [5], provided a sufficient condition on W thatallows to obtain (3.1).THEOREM 3 (Serra&Tilli, [5]). Assume that, for some real number m 0,(3.4) W v 12v 2Hm0 k mλkpk Rn∇kv x pk dx λk 0, pk 1 .Then, assumption (H) is fulfilled if W is the space of those v Hm with ∇kv Lpk0 k m endowed with its natural norm.Moreover, the limit function w obtained via Theorem 2 solves, in the sense ofdistributions, the hyperbolic equation (3.1).REMARK 5. We recall that, as usual, v Hm is the L2 norm of ξ m v ξ , wherev is the Fourier transform of v. The typical case is when m is an integer, so that v Hmreduces to ∇mv L2 .On the extensions of the De Giorgi approach 1573.1. Examples.In addition to (2.1) there are many other second order hyperbolic equations that can beinvestigated using the approach suggested by THEOREM 2&THEOREM 3. We brieflyrecall here some of the most significant ones (for a complete discussion we refer thereader to [5]):1. Nonlinear vibrating–beam equation:w ∆2w ∆pw w q 2w p, q 1 .HereW is defined byW vRn12∆v 2 1p∇v p 1qv q dxand W v H2 : ∇v Lp,v Lq .2. Wave equation with fractional Laplacian:w ∆ s 0 s 1 .HereW is defined byW v cn,sRn Rnv x v y 2x y n 2sdxdy(which is, for a proper choice of cn,s, the natural energy associated to the frac-tional Laplacian) and W Hs.3. Sine–Gordon equation:w ∆w sinw.HereW is defined byW vRn12∇v 2 1 cosv dxand W H1.4. Wave equation with p–Laplacian :w ∆pw.HereW is defined byW v 1p Rn∇v p dxand W v L2 : ∇v Lp .Note that in the cases of the Sine–Gordon and the p–Laplacian equation, thefunctional W satisfies assumption (H), but not assumption (3.4). Consequently, onecould not apply THEOREM 3 to this cases. In the Sine–Gordon case, however, since thefunctional is quadratic in the higher order space derivatives, one can prove an analogousof THEOREM 3. On the contrary, it is an open problem whether this can be done alsofor the case of the p–Laplacian.158 L. Tentarelli4. Extension to nonhomogeneous equations.The natural further extension is the addition of a general forcing term at the right–handside of (3.1), that is, the study of the Cauchy problem associated with the nonhomoge-neous equation(4.1) w t,x ∇W w t, x f t,x .The proper choice for the functional Fε in this case is given by(4.2) Fε ut0e t εRnε2 u t,x2dx W u t,Rnfε t,x u t,x dx dt,where fε is a sequence suitably converging to f .This issue has been the topic of the doctoral dissertation of the author and isextensively investigated in [9]. Here we just announce the result.THEOREM 4 (Tentarelli&Tilli, [9]). LetW be a functional satisfying assump-tion (H) and w0, w1 W. Let also f L2loc 0, ,L2 . Then, there exists a sequencefε , converging to f in L2 0,T ;L2 for all T 0, such that:(a) Minimizers. For every ε 0,1 , the functional Fε defined by (4.2) has a mini-mizers wε in the class of functions in H2loc 0, ;L2 that are subject to (1.2).(b) Estimates. There exist two positive constants Ct ,Cτ,T , depending on t, τ and T(in a continuous way), but independent of ε, such thatRnwε t,x2 dx Ct ,Rnwε t,x 2 dx Ct , t 0,τ TτW wε t, dt Cτ,T , τ 0, T ε,t0wε s2W ds Ct , t 0.(c) Convergence. Every sequence wεi (with εi 0) admits a subsequence which isconvergent in the weak topology of H1 0,T ;L2 , for every T 0, to a functionw that satisfies (1.2) (where the latter is meant as an equality inW ). In addition,w Lloc 0, ;L2 and w L2loc 0, ;W .(d) Energy inequality. Letting E be again the mechanical energy defined by (3.3),there results(4.3) E t E 0 t2t0 Rnf s,x 2 dxds2, for a.e. t 0.(e) Solution of (4.1). Assuming, furthermore, that for some real numbers m0, λk 0 and pk 1,W satisfies (3.4), then the limit function w solves (4.1).On the extensions of the De Giorgi approach 159Some comments are in order. First, we point out that the estimate on the me-chanical energy established by (4.3) is the same that one can find applying a formalGrönwall–type argument to (4.1). In addition, setting f 0, the results of THEOREM4 recover exactly those of THEOREM 2&THEOREM 3, thus showing that our extensionto nonhomogeneous equations is consistent.On the other hand, the Euler–Lagrange equation satisfied by the minimizers ofFε (which is analogous to (1.4)) suggests to work directly with f in place of fε in (4.2).However, this gives rise to several issues in establishing the requested a priori estimateson Fε wε . On the contrary, a proper choice of fε allows one to adapt the De Giorgiapproach under the sole assumption f L2loc 0, ;L2 , which is the usual one in thesearch of solutions of finite energy for (4.1). In particular, the detection of a proper(topology and) “speed of convergence” for fε to f is one of the main issues in theextension to nonhomogeneous problems.Finally, it is worth to outline briefly the main difference between the homo-geneous and the nonhomogeneous case: the estimates on the sequence wε are nolonger global in time. This occurs since the presence of f drops all the uniform boundsdeduced in [5] and allows to establish estimates that are either independent of ε orindependent of t. In particular, the presence of the forcing term entails that the quantityEε t :12 Rnwε t,x2 dxtε 2 e s t ε s t W wε s, dsis not decreasing (as in the homogeneous case) and not even uniformly bounded withrespect to both ε and t. This function, that we call approximate energy, is a formalapproximation of the mechanical energy E and the investigation of its behavior is themain point of our approach, since it provides the a priori estimates on the minimizerswε. Consequently, the fact that it admits only estimates on bounded intervals is thereason for which the inequalities in (b) are no longer global.This transition “from global to local” of the a priori estimates affects the regular-ity of the limit function w, but fortunately does not rule out the possibility of extendingthe De Giorgi approach. Actually, the proofs of the energy inequality, the initial condi-tions and (4.1) do not require any global estimate on the sequence of minimizers (evenin the homogeneous case).Moreover, we point out that the choice of the sequence fε is crucial also for es-tablishing the proper estimate on Eε t ; in particular, for establishing causal estimatesfor a quantity which is a–causal by definition.5. A further extension: dissipative equations.Finally, it worth recalling that [5] also shows that an approach à la De Giorgi is avail-able also for dissipative homogeneous wave equations of the typew t,x ∇W w t, x ∇G w t, x ,160 L. Tentarelliwhere G is a quadratic form defined on a suitable Hilbert space. A typical example isgiven by the Telegraph equationw ∆w w p 2w w p 1(just settingW v Rn12 ∇v1p vp dx and G v 12 Rn v 2 dx).As in the non–dissipative case, also here it is natural to wonder if an extensionto the nonhomogeneous case, namelyw t,x ∇W w t, x ∇G w t, x f t,x ,is possible. The answer is again positive and this issue will be treated in a forthcomingpaper by the author, as well.References[1] DE GIORGI E., Conjectures concerning some evolution problems. A celebration of John F. Nash, Jr.,Duke Math. J. 81 2 (1996), 255–268.[2] DE GIORGI E., Selected Papers, edited by L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda and S.Spagnolo, Springer-Verlag, Berlin 2006.[3] SERRA E., On a conjecture of De Giorgi concerning nonlinear wave equations, Rend. Semin. Mat.Univ. Politec. Torino 70 1 (2012), 85–92.[4] SERRA E. AND TILLI P., Nonlinear wave equations as limits of convex minimization problems: proofof a conjecture by De Giorgi, Ann. of Math. (2) 175 3 (2012), 1551–1574.[5] SERRA E. AND TILLI P., A minimization approach to hyperbolic Cauchy problems, J. Eur. Math. Soc.18 9 (2016), 2019–2044.[6] STEFANELLI U., The De Giorgi conjecture on elliptic regularization, Math. Models Methods Appl.Sci. 21 6 (2011), 1377–1394.[7] STRAUSS W. A., Nonlinear wave equations, CBMS Regional Conference Series in Mathematics, 73,AMS, Providence, RI, 1989.[8] STRUWE M., On uniqueness and stability for supercritical nonlinear wave and Schrödinger equations,Int. Math. Res. Not., Art. ID 76737 (2006), 14 pp.[9] TENTARELLI L. AND TILLI P., The minimization approach to hyperbolic Cauchy problems: an exten-sion to nonhomogeneous equations, preprint arXiv:1709.09111 [math.AP] (2017).AMS Subject Classification: 35L70, 35L71, 35L75, 35L76, 35L90, 35L15, 49J45.Lorenzo Tentarelli,Dipartimento di Matematica e Applicazioni “R. Caccioppoli"MSA, via Cinthia, I-80126 Napoli, ITALYe-mail: lorenzo.tentarelli@unina.itLavoro pervenuto in redazione il 20.12.2016.

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On the extensions of the De Giorgi approach to nonlinear hyperbolic equations

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On the extensions of the De Giorgi approach to nonlinear hyperbolic equations

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