Related papers: Maximal Regularity for Non-Autonomous Second Order…

We establish sharp higher-order H\"older regularity estimates up to the boundary for solutions to equations of the form $\partial_t u-Lu=f(t,x)$ in $I\times\Omega$ where $I\subset\mathbb{R}$, $\Omega\subset\mathbb{R}^n$ and $f$ is H\"older…

We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $…

We consider elliptic problems with nonclassical boundary conditions that contain additional unknown functions on the border of the domain of the elliptic equation and also contain boundary operators of higher orders with respect to the…

In this article we establish fine results on the boundary behavior of solutions to nonlocal equations in $C^{k,\gamma}$ domains which satisfy local Neumann conditions on the boundary. Such solutions typically blow up at the boundary like $v…

We consider an elliptic Kolmogorov equation lambda u - Ku =f in a convex subset C of a separable Hilbert space X. We prove maximal Sobolev regularity of its weak solution, when lambda >0 and f is in L^2(C,nu), where nu is the log-concave…

We consider the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where…

We study the ``hyperboloidal Cauchy problem'' for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behaviour at the boundary, or with polyhomogeneous initial data.…

We prove the local Hadamard well-posedness of the ``good'' Boussinesq equation formulated on the half-line with nonzero Robin boundary conditions. These boundary data involve the Dirichlet and Neumann boundary values as well as the second…

In this note we derive large-scale regularity properties of solutions to second-order linear elliptic equations with random coefficients on the half- space with homogeneous Neumann boundary data; it is a companion to arXiv:1604.02717 in…

We prove boundary regularity and a compactness result for parabolic nonlocal equations of the form $u_t-Iu=f$, where the operator $I$ is not necessarily translation invariant. As a consequence of this and the regularity results for…

In this article, we study a one-dimensional nonlocal quasilinear problem of the form $u_t=a(\Vert u_x\Vert^2)u_{xx}+\nu f(u)$, with Dirichlet boundary conditions on the interval $[0,\pi]$, where $0<m\leq a(s)\leq M$ for all $s\in…

We study continuous dependence of solutions to quasilinear evolution equations of parabolic-type in the framework of maximal $L^p$-regularity. For equations of the form \[ \frac{d\phi}{dt} + A(t,\phi)\phi = f(t,\phi), \] we establish…

In this paper the asymptotic distributions are exactly solved for linearly independent solutions considering problems of the second order and for the coefficients of asymptotic destribution the recurent formulas are obtained. Further, using…

We prove the existence of highest, cusped, periodic travelling-wave solutions with exact and optimal $ \alpha $-H\"older continuity in a class of fractional negative-order dispersive equations of the form \begin{equation*} u_t + (|…

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain $\Omega$ in $ \IR^N$, with the Navier boundary condition $ u=\Delta u =0 $ on $ \pOm$. Here $…

We prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a…

We examine boundary regularity for a fully nonlinear free transmission problem. We argue using approximation methods, comparing the operators driving the problem with a limiting profile. Working natural conditions on the data of the…

This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions $u$ and $v$, and a domain $\Omega$; with $u$ and $v$ being both positive…

In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying…

We examine the fourth order problem $\Delta^2 u = \lambda f(u) $ in $ \Omega$ with $ \Delta u = u =0 $ on $ \partial \Omega$, where $ \lambda > 0$ is a parameter, $ \Omega$ is a bounded domain in $ R^N$ and where $f$ is one of the following…