THE STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY FOR SYSTEMS. 1. Introduction. n 2. u 2 + nw (u) dx = 0. 2

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1 THE STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY FOR SYSTEMS N. D. ALIKAKOS AND A. C. FALIAGAS Abstract. Utilizing stress-energy tensors which allow for a divergencefree formulation, we establish Pohozaev's identity for certain classes of quasilinear systems with variational structure. In his study of the system. Introduction () u W u (u) = 0, R n x u(x) R m ( W where W : R m R and W u :=,, W ), Gui [4] obtained several m relations (identities) that he called Hamiltonian Identities, which imply a certain degree of rigidity for the solutions of () in the whole space. In [4] these identities were also related to the (vector) Pohozaev identity [3, 7] n () u + nw (u) dx = 0. R n In [] one of the authors, by utilizing that () has as a consequence (3) divt = 0 for an appropriate stress-energy tensor (4) T ij = u,i u,j δ ij u + W (u), obtained a monotonicity formula with various corollaries and also noticed that the Hamiltonian identities in [4] can be extracted from the divergence free structure. Since then we realized that stress-energy tensors had been known in the physics literature for a wide class of Lagrangians ([6, 7]) including (4) above. Also several authors have utilized special instances of the divergence-free formulation ([9]). Our purpose in the present note is the derivation of Pohozaev 000 Mathematics Subject Classication. 49N99, 35B08, 35A5. Key words and phrases. Calculus of variations, stress-energy tensor, p-lapacian, minimal surface.

2 STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY identities for systems, by utilizing the stress-energy tensor for the following two classes of Lagrangians (I) (II) L(u, p) = b klrs(u)p kl p rs + W (u) L(u, p) = ϕ( p ) + W (u) (summation convention applies where appropriate) with corresponding tensors (5) T ij = b kjrs (u)u k,i u r,s δ ij b klrs(u)u k,l u r,s + W (u) and (6) T ij = ϕ ( u )u,i u,j δ ij ϕ( u ) + W (u). In either case from the Euler-Lagrange equations we deduce (7) T ij,j = 0, where the notation u,i = x i for partial derivatives has been adopted.. Class I Hypotheses: (H) Symmetry: b klrs = b rskl. (H) Ellipticity: b klrs (u)p kl p rs c p p R m n, u R m where c > 0 is a constant, (8) p = p ij p ij. and b klrs : R m R are C -functions for k, r =,, m and l, s =,, n. Note that when m = n for p ij = ξ i ξ j, ξ = (ξ,, ξ n ) R n, we obtain from (H) (9) b klrs (u)ξ k ξ l ξ r ξ s c ξ 4. Proposition. Let open, smooth set in R n and u C () C () a solution of the system, i =,, n, (0) (b ijkl (u)u k,l ),j b klrs,i(u)u k,l u r,s W,i (u) = 0, u : R m u = a, x where W is a C potential, W (a) = 0 and let x 0. Then the following identity holds true () 0 = n where the notation is as follows u udx + n W (u)dx + (x x 0 ) ν u ads, p u = b klrs (u)p kl p rs and ν is the outward unit normal on.

3 STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY 3 Proof. Following [8] we integrate (x i T ij ),j = T ii over and apply the divergence theorem to obtain () T ii dx = x i T ij ν j ds. From (5) we obtain (3) T ii = n b klrs (u)u k,l u r,s nw (u) = n u u nw (u). Next we treat the integrand on the right of (). For x x = (x ν)ν + (x h)h, where h T x, h =, is a unit tangent vector eld (hence h ν = 0). Therefore we have x i T ij ν j = ((x ν)ν i + (x h)h i )b kjrs (a)u r,s u k,i ν j (x ν)(l 0 + W (a)) where L 0 ( u) = b klrs(a)u k,l u r,s. For x we have W (a) = 0 and k h = u k,ih i = 0. Hence (4) x i T ij ν j = (x ν)(u k,i ν i ν j b kjrs (a)u r,s L 0 ). Now we observe that for x (5) u i,j (x) = ν j (x)u i,k (x)ν k (x), i =,, m; j =,, n. Indeed, if we set z := ν on, i.e. z i = u i,j ν j, then (z i ν j u i,j ) = (z i ν j u i,j )(z i ν j u i,j ) i,j Utilizing (5) in (4) we obtain = z i z i ν j ν j z i ν j u i,j u i,j z i ν j + u i,j u i,j = z i z i z i z i z i z i + u i,j u i,j = u = 0. (6) x i T ij ν j = (x ν) u a, x. Eq. () now follows from () via (3) and (6).

4 STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY 4 3. Class II: p-laplacian, Minimal Surface Equation Here we consider the system (see [] for the scalar case) (7) div(ϕ ( u ) u) W u (u) = 0, where ϕ C (R + ) such that ϕ(0) = 0 and ϕ (s) 0 s 0. Proposition. Let open, smooth set in R n and u C () C () a solution of (8) div(ϕ ( u ) u) W u (u) = 0, u = a, u : R m x where W is a C potential, W (a) = 0 and let x 0. Then the following identity holds true (9) 0 = n where ψ( u )dx + n W (u)dx + (x x 0 ) ν ψ( )ds, (0) ψ(s) = ϕ(s) n sϕ (s) ψ(s) = sϕ (s) ϕ(s). Proof. We proceed as in the proof of Proposition. Integrating (x i T ij ),j = T ii over and applying the divergence theorem, we obtain () T ii dx = x i T ij ν j ds. From (6) we obtain () For x we have and moreover (3) T ii = ϕ ( u ) u n = n ψ( u ) nw (u). ϕ( u ) + W (u) x = (x ν)ν + (x h)h, h T x, h = (h ν = 0) x i T ij ν j = (x ν)ν i T ij ν j + (x h)h i T ij ν j = (x ν)ϕ ( u ) (x ν)l + (x h)ϕ ( u )u k,i h i u k,j ν j (x h)(h ν)l. Since u,i h i = 0 on ( (4) x i T ij ν j = (x ν) ϕ ( u ) ϕ( u ) W (a) ).

5 STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY 5 Recalling that W (a) = 0, = u on we obtain from (4) (5) x i T ij ν j = (x ν) ψ( ), x. Combining (), () and (5) yields (9). 4. Remarks. We notice that for certain choices of ϕ the functions ψ, ψ are non-negative. For example for the minimal surface choice we have and ψ(s) = ϕ(s) = ( + s ) ψ(s) = ( + s s ) 0 n + s s ( + s + s ) = 0. + s + s. Recalling that u = a, from (8), after taking the inner product with u a and integrating over, we obtain via the divergence theorem ϕ ( u ) u dx = W u (u) (u a)dx, equivalently (6) where ϕ( u )dx = W u (u) (u a)dx, (7) ϕ(s) = sϕ (s). From Pohozaev's identity (9), for = R n and 0 as x, suciently fast, we obtain (8) ψ( u )dx = W (u)dx. R n R n At this point we introduce the additional hypothesis (H3) There are constants α, β > 0 such that a < n, β < n and Examples satisfying (H3). α = β = and n >. α sϕ (s) ϕ(s) β s > 0. (i) For ϕ(s) = s (H3) is trivially satised for

6 STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY 6 (ii) For ϕ(s) = ( + s ) we have ϕ (s) = from which we obtain sϕ (s) ϕ(s) = +, + s sϕ (s) ϕ(s) + s and therefore and (H3) is satised for α =, β = and n >. (iii) For the p-laplacian choice ϕ(s) = s r, 0 < r < n, (H3) is satised for α = β = r. Proceeding from (H3) we have by (7) and by the rst of (0) αϕ(s) ϕ(s) βϕ(s) αψ(s) + α n ϕ(s) ϕ(s) βψ(s) + β n ϕ(s), hence αψ(s) ( α n ) ϕ(s), ( β n ) ϕ(s) βψ(s). Letting (9) (i.e. α, β > 0), we obtain + α n = α, + β n = β α ψ(s) ϕ(s) β ψ(s). Therefore, setting s = u in this relation and integrating over R n yield α ψ( u )dx ϕ( u )dx β ψ( u )dx R n R n R n and utilizing (6) and (8) (30) α W (u)dx (u a) W u (u)dx β W (u)dx, R n R n R n hence (3) β W (u)dx (u a) W u (u)dx α W (u)dx. R n R n R n In the case of examples (i), (ii) above we have equality (3) (u a) W u (u)dx = r W (u)dx, R n R n

7 STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY 7 with + r n = r. Generally, however, under (H3), we only have the estimate (33) α R n (u a) W u (u)dx R n W (u)dx β. Finally, we note that in the above derivations we have α β, hence by (9) α β and by (6) R n (u a) W u (u)dx 0. By (30) R n W (u)dx 0. This follows also from (8) and (0): ψ = ϕ n ϕ = ( ϕ n ϕ )ϕ ( β n )ϕ, where the last inequality follows from (H3). References [] N. D. Alikakos, Some basic facts on the system u W u(u) = 0. Proc. Amer. Math. Soc. 39 (0), no., [] L. Caarelli; N. Garofalo; F. Segala, A gradient bound for entire solutions of quasilinear equations and its consequences. Comm. Pure Appl. Math. 47 (994), no., [3] L. C. Evans, Partial Dierential Equations, Second edition, Graduate Studies in Mathematics 9, AMS 00. [4] C. Gui, Hamiltonian Identities for Elliptic Partial Dierential Equations, J. Funct. Anal. 54 (008), no. 4, [5] J. D. Jackson, Classical electrodynamics, Third edition, Wiley 998. [6] L. D. Landau; E. M. Lifschitz, Course of theoretical physics Vol.. Classical eld theory, Fourth edition, Butterworth-Heinemann 980. [7] M. Struwe, Variational methods. Applications to nonlinear partial dierential equations and Hamiltonian systems, Fourth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer 008. [8] R. Schoen, Lecture notes on general relativity, Stanford University 009. [9] E. Sandier; S. Serfaty, Vortices in the magnetic Ginzburg-Landau model, Progress in Nonlinear Dierential Equations and their Applications 70, Birkhäuser 007. Department of Mathematics, University of Athens, Panepistemiopolis, 5784 Athens, Greece address: nalikako@math.uoa.gr Department of Mathematics, University of Athens, Panepistemiopolis, 5784 Athens, Greece address: afaliaga@math.uoa.gr