Related papers: $L^{\infty}$ estimates and uniqueness results for …

The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $u_t - \Delta u + |\nabla u|^q = 0$ in the whole space $R^N$ is investigated for the critical exponent $q = (N+2)/(N+1)$. Convergence towards a…

The weak Harnack inequality for $L^p$-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that H\"older…

We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton-Jacobi equations.

In the class of admissible weak solutions, we prove a weak-strong uniqueness result for the incompressible Euler equations assuming that the symmetric part of the gradient belongs to $L^1_{\rm loc}([0,+\infty);L^{\rm…

In this note we present $L^\infty$-results for problems of the form $A(x,u,Du)=B(x,u,Du)$ in $\Omega$, $u>0$ in $\Omega$, $u=0$ on $\partial\Omega$, where the growth condition for the function $B\colon \Omega \times \mathbb{R}\times…

We obtain the comparison principle for discontinuous viscosity sub- and supersolutions of nonlocal Hamilton-Jacobi equations, with superlinear and coercive gradient terms. The nonlocal terms are integro-differential operators in L\'evy…

We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } \Omega $$…

We reduce the problem of proving decay estimates for viscosity solutions of fully nonlinear PDEs to proving analogous estimates for solutions of one-dimensional ordinary differential inequalities. Our machinery allow the ellipticity to…

Let $H$ be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation $U_{t}+H(U_x)=0$ and signed Radon measure valued entropy solutions of the conservation law…

In this paper, we will study the following parabolic problem $u_t - div(\omega(x) \nabla u)= h(t) f(u) + l(t) g(u)$ with non-negative initial conditions pertaining to $C_b(\mathbb{R}^N)$, where the weight $\omega$ is an appropriate function…

This article is devoted to the optimal control of state equations with memory of the form: ?[x(t) = F(x(t),u(t), \int_0^{+\infty} A(s) x(t-s) ds), t>0, with initial conditions x(0)=x, x(-s)=z(s), s>0.]Denoting by $y_{x,z,u}$ the solution of…

The aim of this paper is twofold. - In the setting of RCD(K,$\infty$) metric measure spaces, we derive uniform gradient and Laplacian contraction estimates along solutions of the viscous approximation of the Hamilton--Jacobi equation. We…

We establish both the existence and uniqueness of non-negative global solutions for the nonlinear heat equation $u_t-\Delta u=|x|^{-\gamma}\,u^q$, $0<q<1$, $\gamma>0$ in the whole space $\mathbb{R}^N$, and for non-negative initial data…

Here, we consider anisotropic degenerate parabolic-hyperbolic equations and degenerate quasilinear Hamilton-Jacobi equations. We prove the equivalence of two notions of entropy and viscosity solutions of two equations, and apply it to…

This paper deals with the periodic homogenization of nonlocal parabolic Hamilton-Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal…

We consider the Hamilton-Jacobi equation \[{H}(x,u,Du)=0,\quad x\in M, \] where $M$ is a connected, closed and smooth Riemannian manifold, ${H}(x,u,p)$ satisfies Tonelli conditions with respect to $p$ and certain decreasing condition with…

We study nonlocal first-order equations arising in the theory of dislocations. We prove the existence and uniqueness of the solutions of these equations in the case of positive and negative velocities, under suitable regularity assumptions…

We consider perturbations of the diffusive Hamilton-Jacobi equation \begin{equation*} %\label{non_pert} \left\{ \begin{array}{lcl} \hfill -\Delta u &=& (1+g(x))| \nabla u|^p\qquad \mbox{ in } \IR^N_+, \\ \hfill u &=& 0 \hfill \mbox{ on }…

The present paper first aims to study the BV-type regularity for viscosity solutions of the Hamilton-Jacobi equation \[ u_t(t,x)+H\big(D_{x} u(t,x)\big)~=~0\qquad\forall (t,x)\in ]0,\infty[\times\mathbb{R}^d \] with a coercive and uniformly…

We consider non-negative, weak solutions to the doubly nonlinear parabolic equation $$ \partial_t u^q-\mbox{div}(|Du|^{p-2}Du)=0 $$ in the super-critical fast diffusion regime $0<p-1<q<\frac{N(p-1)}{(N-p)_+}$. We show that when solutions…