On the extensions of the De Giorgi approach to nonlinear hyperbolic equations
aa r X i v : . [ m a t h . A P ] A p r ON THE EXTENSIONS OF THE DE GIORGI APPROACHTO NONLINEAR HYPERBOLIC EQUATIONS
LORENZO TENTARELLI Abstract.
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 nonhomogeneousequations, obtained by the author in [9] in collaboration with P. Tilli.
Contents
1. De Giorgi’s conjecture. 12. The proof of the conjecture. 33. Extension to homogeneous equations. 43.1. Examples. 64. Extension to nonhomogeneous equations. 75. A further extension: dissipative equations. 8References 91.
De Giorgi’s conjecture.
In 1996, E. De Giorgi stated the following conjecture on weak solutions of the defocusing
NLWequation.
Conjecture 1.1 (De Giorgi, [1, 2]) . Let w , w ∈ C ∞ ( R n ) , let k > be an integer; for everypositive real number ε , let w ε = w ε ( t, x ) be the minimizer of the functional F ε ( u ) := Z ∞ Z R n e − t/ε (cid:16) | u ′′ ( t, x ) | + ε |∇ u ( t, x ) | + ε | u ( t, x ) | k (cid:17) dx dt (1) in the class of all u satisfying the initial conditions u (0 , x ) = w ( x ) , u ′ (0 , x ) = w ( x ) . (2) Then, there exists lim ε ↓ w ε ( t, x ) = w ( t, x ) , satisfying the equation w ′′ = ∆ w − kw k − . (3) Remark . In the statement of the conjecture, we maintained the original formulation of [1, 2]and we only changed notation, according to that we use in the sequel. The same thing holds forall the results we mention in this paper. In addition, we recall that u ′ ( t, x ) denotes ∂u∂t ( t, x ) and Author supported by the FIR grant 2013 “Condensed Matter in Mathematical Physics (Cond-Math)” (codeRBFR 13WAET). that, for the sake of simplicity, we always omit the dependence of the functional spaces on R n ,i.e. H = H ( R n ), L p = L p ( R n ) and so on.In order to better understand the meaning of the conjecture, it is worth stressing some char-acteristic features of the functional F ε .First we note that it involves second order time derivatives. Thus, a minimizer of F ε solves afourth order PDE. However, if one computes the formal Euler–Lagrange equation satisfied by aminimizer w ε , then one obtains ε ( e − t/ε w ′′ ε ) ′′ = e − t/ε (∆ w ε − kw k − ε )and thus, expanding and dropping e − t/ε , ε w ′′′′ ε − εw ′′′ ε + w ′′ ε = ∆ w ε − kw k − ε . (4)Consequently, if one assumes that w ε → w in some suitable sense and lets ε ↓
0, then oneformally obtains (3).On the other hand, we also remark that, as F ε is defined through integrals over the “space–time” [0 , ∞ ) × R n , the initial conditions of the Cauchy problem are in fact boundary conditions for the minimization problem.In addition, it is convenient to stress the singular nature of the integration weight e − t/ε . Moreprecisely, one can see that ε − e − t/ε dt is an approximate Dirac delta measure and hence, at leastformally, εF ε ( u ) ≈ Z R n (cid:0) |∇ w ( x ) | + | w ( x ) | k (cid:1) dx, as ε ↓ . Hence, this prevents a straightforward application of classical techniques of variational con-vergence, such as Γ– convergence . The previous asymptotic expansion, indeed, shows that thistechnique does not provide useful information on the limit behavior of the sequence of theminimizers.Finally, one can note that F ε is convex (for fixed ε >
0) and that therefore, up to somesuitable technical adaptation, the proof of the existence and uniqueness of the minimizers is nota demanding issue.We also recall that the existence of global solutions for the Cauchy problem (3)&(2) is notnew (see e.g. [7] and the references therein). Actually, as highlighted in [3, 4], the originality ofthe strategy hinted by De Giorgi lies in how he intended to exploit techniques from the Calculusof Variations. The variational approaches to the wave equation w ′′ = ∆ w and its nonlinearvariants that can one can find in the literature (see e.g. [7, 8] and references therein) are basedon the interpretation of w ′′ = ∆ w as the Euler–Lagrange equation of the functional I ( w ) := Z ∞ Z R n (cid:0) | w ′ ( t, x ) | − |∇ w ( t, x ) | (cid:1) dx dt (with possibly lower order terms like | w | k ). However, since I is neither convex nor bounded frombelow, one is forced to search for critical points rather than global minimizers . Unfortunately,functionals like I behave badly also for the application of Critical Point Theory, so that onlypartial results can be proved. De Giorgi, on the contrary, introduces a new functional F ε thatis quite easy to minimize (regardless of the magnitude of k ) and thus moves the problem to theinvestigation of the limit behavior of the sequence of the minimizers. N THE EXTENSIONS OF THE DE GIORGI APPROACH 3 The proof of the conjecture.
In 2012, E. Serra and P. Tilli showed that Conjecture 1.1 is in fact true. Precisely, in [4], theyproved the following theorem.
Theorem 2.1 (Serra&Tilli, [4]) . For p ≥ and ε > , let w ε ( t, x ) denote the unique minimizerof the strictly convex functional F ε ( u ) = Z ∞ Z R n e − t/ε (cid:0) | u ′′ ( t, x ) | + ε |∇ u ( t, x ) | + ε | u ( t, x ) | p (cid:1) dx dt under the boundary conditions (2) , where w and w are given functions such that w , w ∈ H ∩ L p . Then: (a)
Estimates. There exists a constant C (which depends only on w , w , p and n ) suchthat, for every ε ∈ (0 , , Z T Z R n (cid:0) |∇ w ε ( t, x ) | + | w ε ( t, x ) | p (cid:1) dx dt ≤ CT, ∀ T > ε, Z R n | w ′ ε ( t, x ) | dx ≤ C and Z R n | w ε ( t, x ) | dx ≤ C (1 + t ) , ∀ t ≥ , and, for every function h ∈ H ∩ L p (cid:12)(cid:12)(cid:12)(cid:12)Z R n w ′′ ε ( t, x ) h ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( k h k L p + k∇ h k L ) , for a.e. t > . (b) Convergence. Every sequence w ε i (with ε i ↓ ) admits a subsequence which is convergent,in the strong topology of L q ((0 , T ) × A ) for every T > and every bounded open set A ⊂ R n (with arbitrary q ∈ [2 , p ) if p > and q = p if p = 2 ), almost everywhere in R + × R n and in the weak topology of H ((0 , T ) × R n ) for every T > , to a function w such that w ∈ L ∞ ( R + ; L p ) , ∇ w ∈ L ∞ ( R + ; L ) ,w ′ ∈ L ∞ ( R + ; L ) , w ∈ L ∞ ((0 , T ); H ) ∀ T > , which solves in R + × R n the nonlinear wave equation w ′′ = ∆ w − p | w | p − w (5) with initial conditions as in (2) . (c) Energy inequality. Letting E ( t ) := Z R n (cid:0) | w ′ ( t, x ) | + |∇ w ( t, x ) | + | w ( t, x ) | p (cid:1) dx, the function w ( t, x ) satisfies the energy inequality E ( t ) ≤ E (0) = Z R n (cid:0) | w ( x ) | + |∇ w ( x ) | + | w ( x ) | p (cid:1) dx, for a.e. t > . L. TENTARELLI
Remark . We stress the fact that, in (b) , the limit function w solves (5) in a distributional (or weak ) sense, namely Z ∞ Z R n w ′ ( t, x ) ϕ ′ ( t, x ) dx dt = Z ∞ Z R n ∇ w ( t, x ) · ∇ ϕ ( t, x ) dx dt ++ Z ∞ Z R n p | w ( t, x ) | p − w ( t, x ) ϕ ( t, x ) dx dt for every ϕ ∈ C ∞ ( R + × R n ). In the sequel we only deal with this type of solutions.Some comments are in order. First, Conjecture | w | k with k integer , while Theorem | w | p without the assumption of p integer. Another relevantfeature of Theorem w , w are much weakerthan those of the conjecture.On the other hand, the convergence of the sequence of the minimizers is obtained up toextracting subsequences, thus “losing” the uniqueness claimed in the conjecture. In particular,it is an open problem to avoid the extraction of subsequences when p is large.In addition, Theorem mechanical energy E usually as-sociated with (5), which proves that the obtained solutions are of energy class in the sense ofStruwe (see [8]). When p is “sufficiently” small, the inequality is in 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 version of the conjec-ture on bounded intervals. However, that paper only deals with the proof of (5) and does nottreat the fulfillment of the initial condition w ′ (0 , x ) = w ( x ).3. Extension to homogeneous equations.
Now, one can easily see that, setting W ( v ) = Z R n (cid:18) |∇ v | + 1 p | v | p (cid:19) dx, up to some multiplicative constants equation (5) reads w ′′ ( t, x ) = −∇W ( w ( t, · ))( x ) , (6)where ∇W denotes the Gˆateaux derivative of the functional W . Therefore, it is natural towonder if the sequence of the minimizers of the functional F ε , that here is defined by F ε ( u ) := Z t e − t/ε (cid:18) Z R n ε | u ′′ ( t, x ) | dx + W ( u ( t, · )) (cid:19) dt, (7)converges to a solution of the Cauchy problem associated with (6), even for different choices of W . Remark . In (7) one uses a different scaling in ε , with respect to (1). This is due to the factthat in the abstract framework this choice simplifies computations. However, this does not yieldsignificant differences.This problem has been solved again by E. Serra and P. Tilli, in [5]. Before showing thestatements of the main results, it is necessary to point out under which assumptions on thefunctional W (that we refer to as assumption (H) in the following), they are valid. N THE EXTENSIONS OF THE DE GIORGI APPROACH 5 (H)
The functional W : L → [0 , ∞ ] is lower semi–continuous in the weak topology of L , i.e W ( v ) ≤ lim inf k W ( v k ) , whenever v k ⇀ v in L . Moreover, we assume that the set of functionsW = { v ∈ L : W ( v ) < ∞} is a Banach space such that C ∞ ֒ → W ֒ → L (dense embeddings).Finally, W is Gˆateaux differentiable on W and its derivative ∇W : W → W ′ satisfies k∇W ( v ) k W ′ ≤ C (1 + W ( v ) θ ) , ∀ v ∈ W , for suitable constants C ≥ θ ∈ (0 , Remark . Assumption (H) is typically satisfied by standard functionals like W ( v ) = 1 p Z R n |∇ k v | p dx, p > , (with possibly lower order terms) where W is the space of the L functions v with ∇ k v ∈ L p . Theorem 3.1 (Serra&Tilli, [5]) . Given w , w ∈ W and ε ∈ (0 , , under assumption (H) thefunctional F ε defined in (7) has a minimizer w ε in the space H loc ([0 , ∞ ); L ) subject to (2) .Moreover: (a) Estimates. There exists a constant C, independent of ε , such that Z τ + Tτ W ( w ε ( t, · )) dt ≤ CT, ∀ τ ≥ , ∀ T ≥ ε, Z R n | w ′ ε ( t, x ) | dx ≤ C and Z R n | w ε ( t, x ) | dx ≤ C (1 + t ) , ∀ t ≥ , k w ε k L ∞ ( R + ;W ′ ) ≤ C. (b) Convergence. Every sequence w ε i (with ε i ↓ ) admits a subsequence which is convergent,in the weak topology of H ((0 , T ); L ) for every T > , to a function w such that w ∈ H loc ([0 , ∞ ); L ) , w ′ ∈ L ∞ ( R + ; L ) , w ′′ ∈ L ∞ ( R + ; W ′ ) . Moreover, w satisfies the initial conditions (2) . (c) Energy inequality. Letting E ( t ) := 12 Z R n | w ′ ( t, x ) | dx + W ( w ( t, · )) , (8) the function w ( t, x ) satisfies the energy inequality E ( t ) ≤ E (0) = 12 Z R n | w ( x ) | dx + W ( w ) for a.e. t > . Unfortunately, under these assumptions, it is not known whether w satisfies (6). Anyway,Serra&Tilli, still in [5], provided a sufficient condition on W that allows to obtain (6). L. TENTARELLI
Theorem 3.2 (Serra&Tilli, [5]) . Assume that, for some real number m > , W ( v ) = 12 k v k H m + X ≤ k
In addition to (5) there are many other second order hyperbolic equations thatcan be investigated using the approach suggested by
Theorem
Theorem
Nonlinear vibrating–beam equation : w ′′ = − ∆ w + ∆ p w − | w | q − w ( p, q > . Here W is defined by W ( v ) = Z R n (cid:18) | ∆ v | + 1 p |∇ v | p + 1 q | v | q (cid:19) dx and W = { v ∈ H : ∇ v ∈ L p , v ∈ L q } .2. Wave equation with fractional Laplacian : w ′′ = − ( − ∆) s (0 < s < . Here W is defined by W ( v ) = c n,s Z R n × R n | v ( x ) − v ( y ) | | x − y | n +2 s dx dy (which is, for a proper choice of c n,s , the natural energy associated to the fractionalLaplacian) and W = H s .3. Sine–Gordon equation : w ′′ = ∆ w − sin w. Here W is defined by W ( v ) = Z R n (cid:18) |∇ v | + 1 − cos v (cid:19) dx and W = H .4. Wave equation with p –Laplacian : w ′′ = ∆ p w. Here W is defined by W ( v ) = 1 p Z R n |∇ v | p dx and W = { v ∈ L : ∇ v ∈ L p } . N THE EXTENSIONS OF THE DE GIORGI APPROACH 7
Note that in the cases of the Sine–Gordon and the p –Laplacian equation, the functional W satisfies assumption (H) , but not assumption (9). Consequently, one could not apply Theorem
Theorem p –Laplacian.4. Extension to nonhomogeneous equations.
The natural further extension is the addition of a general forcing term at the right–hand sideof (6), that is, the study of the Cauchy problem associated with the nonhomogeneous equation w ′′ ( t, x ) = −∇W ( w ( t, · ))( x ) + f ( t, x ) . (10)The proper choice for the functional F ε in this case is given by F ε ( u ) = Z t e − t/ε (cid:18) Z R n ε | u ′′ ( t, x ) | dx + W ( u ( t, · )) − Z R n f ε ( t, x ) u ( t, x ) dx (cid:19) dt, (11)where ( f ε ) is a sequence suitably converging to f .This issue has been the topic of the doctoral dissertation of the author and is extensivelyinvestigated in [9]. Here we just announce the result. Theorem 4.1 (Tentarelli&Tilli, [9]) . Let W be a functional satisfying assumption (H) and w , w ∈ W . Let also f ∈ L loc ([0 , ∞ ) , L ) . Then, there exists a sequence ( f ε ) , converging to f in L ([0 , T ]; L ) for all T > , such that: (a) Minimizers. For every ε ∈ (0 , , the functional F ε defined by (11) has a minimizers w ε in the class of functions in H loc ([0 , ∞ ); L ) that are subject to (2) . (b) Estimates. There exist two positive constants C t , C τ,T , depending on t, τ and T (in acontinuous way), but independent of ε , such that Z R n | w ′ ε ( t, x ) | dx ≤ C t , Z R n | w ε ( t, x ) | dx ≤ C t , ∀ t ≥ , Z τ + Tτ W ( w ε ( t, · )) dt ≤ C τ,T , ∀ τ ≥ , ∀ T > ε, Z t k w ′′ ε ( s ) k ′ ds ≤ C t , ∀ t ≥ . (c) Convergence. Every sequence w ε i (with ε i ↓ ) admits a subsequence which is convergentin the weak topology of H ([0 , T ]; L ) , for every T > , to a function w that satisfies (2) (where the latter is meant as an equality in W ′ ). In addition, w ′ ∈ L ∞ loc ([0 , ∞ ); L ) and w ′′ ∈ L loc ([0 , ∞ ); W ′ ) . (d) Energy inequality. Letting E be again the mechanical energy defined by (8) , there results E ( t ) ≤ p E (0) + s t Z t Z R n | f ( s, x ) | dx ds , for a.e. t ≥ . (12)(e) Solution of (10) . Assuming, furthermore, that for some real numbers m > , λ k ≥ and p k > , W satisfies (9) , then the limit function w solves (10) . L. TENTARELLI
Some comments are in order. First, we point out that the estimate on the mechanical energyestablished by (12) is the same that one can find applying a formal
Gr¨onwall–type argument to(10). In addition, setting f ≡
0, the results of
Theorem
Theorem
Theorem F ε (which isanalogous to (4)) suggests to work directly with f in place of f ε in (11). However, this givesrise to several issues in establishing the requested a priori estimates on F ε ( w ε ). On the contrary,a proper choice of ( f ε ) allows one to adapt the De Giorgi approach under the sole assumption f ∈ L loc ([0 , ∞ ); L ), which is the usual one in the search of solutions of finite energy for (10).In particular, the detection of a proper (topology and) “speed of convergence” for f ε to f is oneof the main issues in the extension to nonhomogeneous problems.Finally, it is worth to outline briefly the main difference between the homogeneous and thenonhomogeneous case: the estimates on the sequence ( w ε ) are no longer global in time. Thisoccurs since the presence of f drops all the uniform bounds deduced in [5] and allows to establishestimates that are either independent of ε or independent of t . In particular, the presence of theforcing term entails that the quantity E ε ( t ) := 12 Z R n | w ′ ε ( t, x ) | dx + Z ∞ t ε − e − ( s − t ) /ε ( s − t ) W ( w ε ( s, · )) ds is not decreasing (as in the homogeneous case) and not even uniformly bounded with respectto both ε and t . This function, that we call approximate energy , is a formal approximation ofthe mechanical energy E and the investigation of its behavior is the main point of our approach,since it provides the a priori estimates on the minimizers w ε . Consequently, the fact that itadmits only estimates on bounded intervals is the reason for which the inequalities in (b) are nolonger global.This transition “from global to local” of the a priori estimates affects the regularity of thelimit function w , but fortunately does not rule out the possibility of extending the De Giorgiapproach. Actually, the proofs of the energy inequality, the initial conditions and (10) do notrequire any global estimate on the sequence of minimizers (even in the homogeneous case).Moreover, we point out that the choice of the sequence ( f ε ) is crucial also for establishing theproper estimate on E ε ( t ); in particular, for establishing causal estimates for 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 `a la De Giorgi is available alsofor dissipative homogeneous wave equations of the type w ′′ ( t, x ) = −∇W ( w ( t, · ))( x ) − ∇G ( w ′ ( t, · ))( x ) , where G is a quadratic form defined on a suitable Hilbert space. A typical example is given bythe Telegraph equation w ′′ = ∆ w − | w | p − w − w ′ ( p > W ( v ) = R R n (cid:16) |∇ v | + p | v | p (cid:17) dx and G ( v ) = R R n | v | dx ).As in the non–dissipative case, also here it is natural to wonder if an extension to the nonho-mogeneous case, namely w ′′ ( t, x ) = −∇W ( w ( t, · ))( x ) − ∇G ( w ′ ( t, · ))( x ) + f ( t, x ) , N THE EXTENSIONS OF THE DE GIORGI APPROACH 9 is possible. The answer is again positive and this issue will be treated in a forthcoming paperby the author, as well.
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J. Eur. Math. Soc. (2016),no. 9, 2019–2044.[6] U. Stefanelli, The De Giorgi conjecture on elliptic regularization, Math. Models Methods Appl. Sci. (2011),no. 6, 1377–1394.[7] W.A. Strauss, Nonlinear wave equations , CBMS Regional Conference Series in Mathematics, 73, AMS,Providence, RI, 1989.[8] M. Struwe, On uniqueness and stability for supercritical nonlinear wave and Schr¨odinger equations,
Int.Math. Res. Not. (2006), Art. ID 76737 , 14 pp.[9] L. Tentarelli, P. Tilli, The minimization approach to hyperbolic Cauchy problems: an extension to nonho-mogeneous equations, preprint , arXiv:1709.09111 [math.AP] (2017).
Sapienza Universit`a di Roma, Dipartimento di Matematica, Piazzale Aldo Moro, 5, 00185, Roma,Italy.
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