We prove the existence of a principal eigenvalue associated to the ∞-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.

We prove the existence of a principal eigenvalue associated to the ∞-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.

We examine the regularity of weak and very weak solutions of the Poisson equation on polygonal domains with data in L². We consider mixed Dirichlet, Neumann and Robin boundary conditions. We also describe the singular part of weak and very weak solutions.

Let $X$ be a submanifold of a manifold $Z$. We address the question: When do viscosity subsolutions of a fully nonlinear PDE on $Z$, restrict to be viscosity subsolutions of the restricted subequation on $X$? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be transformed...

The aim of this paper is to establish the existence of at least three solutions for the nonlinear Neumann boundary-value problem involving the p(x)-Laplacian of the form $-{\Delta }_{p\left(x\right)}u+a\left(x\right)|u^{\left|p\right(x)-2}u=\mu g(x,u)$ in Ω, ${\left|\nabla u\right|}^{p\left(x\right)-2}\partial u/\partial \nu =\lambda f(x,u)$ on ∂Ω. Our technical approach is based on the three critical points theorem due to Ricceri.

We prove that strong solutions of the Boussinesq equations in 2D and 3D can be extended as analytic functions of complex time. As a consequence we obtain the backward uniqueness of solutions.

We show time regularity of weak solutions for unsteady motion equations of generalized Newtonian fluids described by $p(x,t)$-power law for $p(x,t)\ge (3n+2)/(n+2)$, $n\ge 2,$ by using a higher integrability property and fractional difference method. Moreover, as its application we prove that every weak solution to the problem becomes a local in time strong solution and that it is unique.

We construct travelling wave graphs of the form $z=-ct+\phi \left(x\right)$, $\phi :x\in {ℝ}^{N-1}\mapsto \phi \left(x\right)\in ℝ$, $N\ge 2$, solutions to the $N$-dimensional forced mean curvature motion $V_n=-c_0+\kappa$ ($c\ge c_0$) with prescribed asymptotics. For any $1$-homogeneous function ${\phi }_{\infty }$, viscosity solution to the eikonal equation $|D{\phi }_{\infty }|=\sqrt{{(c/c_0)}^2-1}$, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by ${\phi }_{\infty }$. We also describe ${\phi }_{\infty }$ in terms of a probability measure on ${\S }^{N-2}$.

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data $u_0\in X$, where $X\in \{M_{2,q}^s\left(ℝ\right),H^{\sigma }\left(𝕋\right),H^{s_1}\left(ℝ\right)+H^{s_2}\left(𝕋\right)\}$ and $q\in [1,2]$, $s\ge 0$, or $\sigma \ge 0$, or $s_2\ge s_1\ge 0$. Moreover, if $M_{2,q}^s\left(ℝ\right)\hookrightarrow L^3\left(ℝ\right)$, or if $\sigma \ge \frac{1}{6}$, or if $s_1\ge \frac{1}{6}$ and $s_2>\frac{1}{2}$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal...

We consider a class of stationary viscous Hamilton-Jacobi equations aswhere $\lambda \ge 0$, $A\left(x\right)$ is a bounded and uniformly elliptic matrix and $H(x,\xi )$ is convex in $\xi$ and grows at most like ${\left|\xi \right|}^q+f\left(x\right)$, with $1\<q\<2$ and $f\in L^{N/q^{\text{'}}}\left(\Omega \right)$. Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate,i.e.${(1+|u\left|\right)}^{\overline{q}-1}u\in H_0^1\left(\Omega \right)$, for a certain (optimal) exponent $\overline{q}$. This completes the recent results in [15],...

We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $-{\sum }_{i,j=1}^n{D}_j\left(a_{ij}\left(x\right)D_{i}u\left(x\right)\right)+b\left(x\right)u\left(x\right)+div\left(\Phi \left(u\left(x\right)\right)\right)=g\left(x\right)-{\sum }_{j=1}^n{f}_j\left(x\right)$ on Ω in the setting of the space H₀(Ω).

Consider the Navier-Stokes equation with the initial data $a\in L_{\sigma }^2\left({ℝ}^d\right)$. Let $u$ and $v$ be two weak solutions with the same initial value $a$. If $u$ satisfies the usual energy inequality and if $\nabla v\in L^2((0,T);{\dot{X}}_1{\left({ℝ}^d\right)}^d)$ where ${\dot{X}}_1\left({ℝ}^d\right)$ is the multiplier space, then we have $u=v$.

A generalized approach to several universality results is given by replacing holomorphic or harmonic functions by zero solutions of arbitrary linear partial differential operators. Instead of the approximation theorems of Runge and others, we use an approximation theorem of Hörmander.

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.

We consider the theory of very weak solutions of the stationary Stokes system with nonhomogeneous boundary data and divergence in domains of half space type, such as ${ℝ}_{+}^n$, bent half spaces whose boundary can be written as the graph of a Lipschitz function, perturbed half spaces as local but possibly large perturbations of ${ℝ}_{+}^n$, and in aperture domains. The proofs are based on duality arguments and corresponding results for strong solutions in these domains, which have to be constructed in homogeneous...