A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton–Jacobi equations

Abstract

We consider the elliptic and parabolic superquadratic diffusive Hamilton-Jacobi equations: ∆u + |∇u| p = 0 and ut = ∆u + |∇u| p , with p > 2 and homogeneous Dirichlet conditions. For the elliptic problem in a half-space, we prove a Liouville-type classification, or symmetry result, which asserts that any solution has to be one-dimensional. This turns out to be an efficient tool to study the behavior of boundary gradient blow-up (GBU) solutions of the parabolic problem in general bounded domains of R n with smooth boundaries.Namely, we show that in a neighborhood of the boundary, at leading order, solutions display a global ODE type behavior of the form uνν ∼ −u p ν , with domination of the normal derivatives upon the tangential derivatives. This leads to the existence of a universal, sharp blow-up profile in the normal direction at any GBU point, and moreover implies that the behavior in the tangential direction is more singular. A description of the space-time profile is also obtained. The ODE type behavior and its connection with the Liouvilletype theorem can be considered as an analogue of the well-known results of Merle and Zaag [31] for the subcritical semilinear heat equation, with the significant difference that for the latter, the ODE behavior is in the time direction (instead of the normal spatial direction).On the other hand, it is known that any GBU solution admits a weak continuation, under the form of a global viscosity solution. As another consequence, we show that these viscosity solutions generically lose boundary conditions after GBU. Namely, solutions without loss of boundary conditions after GBU are exceptional and can be characterized as thresholds between global classical solutions and GBU solutions which lose boundary conditions. This result, as well as the above GBU profile, were up to now essentially known only in one space-dimension.Finally, in the case of elliptic Dirichlet problems, we deduce from our Liouville theorem an optimal Bernstein-type estimate, which gives a partial improvement of a local estimate of P.-L. Lions [28].

A note on one-dimensional symmetry for Hamilton-Jacobi equations with extremal Pucci operators and application to Bernstein type estimate

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We prove a Liouville-type theorem that is one-dimensional symmetry and classification results for non-negative L q -viscosity solutions of the equation, where M ± λ,Λ are the Pucci's operators with parameters λ, Λ ∈ R+ 0 < λ ≤ Λ and p > 1. The results are an extension of the results by Porreta and Verón in [29] for the case p ∈ (1, 2] and by o Filippucci, Pucci and Souplet in [23] for the case p > 2, both for the Laplacian case (i.e. λ = Λ = 1).

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“…, where M ± λ,Λ are the Pucci's operators with parameters λ, Λ ∈ R+ 0 < λ ≤ Λ and p > 1. The results are an extension of the results by Porreta and Verón in [29] for the case p ∈ (1, 2] and by o Filippucci, Pucci and Souplet in [23] for the case p > 2, both for the Laplacian case (i.e. λ = Λ = 1).…”

Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem

“…Theorem 1.1 belongs to the large family of symmetry results for solutions of nonlinear elliptic equations. Besides [4], we just mention the works [12,11,10], and we refer to the references therein, as recent symmetry results for solutions of semilinear, quasilinear and fully nonlinear equations respectively. Moreover, we refer to [3] for symmetry results in halfspaces for nonlocal operators.…”

A nonlinear elliptic problem involving the gradient on a half space

“…Let us finally mention that for the whole space case, it was proved in [37] that any classical solution of (4) when Ω = R N with p > 1 has to be constant. Also, for the half-space problem (1) in the superquadratic case p > 2, it was proved in [27] a Liouville-type classification, or symmetry result, which asserts that any solution…”

“…We also mention the recent work [27] where they examine various results, some of which are Liouville theorems related to (1). 1.2.…”