Related papers: Non-Autonomous Maximal Regularity for Forms of Bou…

Supposing only that $\displaystyle\lim_{t \to 0} \frac{f(t)}{t} = 0$ and $\displaystyle\lim_{t \to \infty} \frac{f(t)}{t^{p}} = 0$, for some $p \in \left(1,\frac{N+1}{N-1}\right)$, we prove that solutions to the extension problem…

We study admissible observation operators for perturbed evolution equations using the concept of maximal regularity. We first show the invariance of the maximal $L^p$-regularity under non-autonomous Miyadera-Voigt perturbations. Second, we…

Motivated by an influential result of Bourgain and Tzafriri, we consider continuous matrix functions $A:\mathbb{R}\to M_{n\times n}$ and lower $\ell_2$-norm bounds associated with their restriction to certain subspaces. We prove that for…

We consider a nonlinear control system depending on two controls u and v, with dynamics affine in the (unbounded) derivative of u, and v appearing initially only in the drift term. Recently, motivated by applications to optimization…

It is known that for a possibly degenerate hypoelliptic Ornstein-Uhlenbeck operator $$ L= \frac{1}{2}\text{ tr} (QD^2 ) + \langle Ax, D \rangle = \frac{1}{2}\text{ div} (Q D ) + \langle Ax, D \rangle,\;\; x \in R^N, $$ all (globally)…

In this paper we study the existence of solutions for nonlinear boundary value problems ({\phi}(u' ))' = f(t,u,u'), l(u,u')=0 where l(u,u') =0 denotes the Dirichlet or mixed conditions on [0, T], {\phi} is a bounded, singular or classic…

Consider the parabolic free boundary problem $$ \Delta u - \partial_t u = 0 \textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} . $$ For a realistic class of solutions, containing for example {\em all} limits of the singular…

Given a class of nonautonomous elliptic operators $\A(t)$ with unbounded coefficients, defined in $\overline{I \times \Om}$ (where $I$ is a right-halfline or $I=\R$ and $\Om\subset \Rd$ is possibly unbounded), we prove existence and…

Inspired by [6, 7], we study the boundary regularity of constant curvature hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$, which have prescribed asymptotic boundary at infinity. Through constructing the boundary expansions of the…

We establish the optimal $C_{H}^{1,1}$ interior regularity of solutions to \[ \Delta_{H}u=f\chi_{\{u\ne0\}}, \] where $\Delta_{H}$ denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on $f$,…

We consider fractional operators of the form $$\mathcal{H}^s=(\partial_t -\mathrm{div}_{x} ( A(x,t)\nabla_{x}))^s,\ (x,t)\in\mathbb R^n\times\mathbb R,$$ where $s\in (0,1)$ and $A=A(x,t)=\{A_{i,j}(x,t)\}_{i,j=1}^{n}$ is an accretive,…

This paper is concerned with the existence and decay of solutions of the following Timoshenko system: $$ \left\|\begin{array}{cc} u"-\mu(t)\Delta u+\alpha_1 \displaystyle\sum_{i=1}^{n}\frac{\partial v}{\partial x_{i}}=0,\, \in \Omega\times…

We establish the optimal regularity of solutions to the Neumann problem for the fractional Laplacian, $(-\Delta)^s u=h$ in $\Omega$, with the external condition $\mathcal N^s u=0$ in $\Omega^c$. For this, a key point is to establish a 1D…

We study the regularity of the bounded self-similar solution to the one-phase Stefan problem with fractional diffusion posed on the whole line. In terms of the enthalpy $h(x,t)$, the evolution problem reads \[ \begin{cases} \partial_t h +…

This paper deals with a problem which describes tuberculosis granuloma formation \begin{align*} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) - uv - u + \beta, &x \in \Omega,\ t>0, \\ v_t = \Delta v + v -uv + \mu w, &x \in…

Let $\Omega\subseteq\mathbb{R}^{d}$ be open, $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^{\infty}$ coefficients, $b$ and $c$ two $d$-dimensional vector-valued functions on $\Omega$ with…

Let L be a second order, uniformly elliptic operator, and consider the equation L u=f under the homogeneous boundary condition u=0. It is well known that f in C(Om) (Om= Omega) does not guarantee second order derivatives D^2 u in C(Om).…

This paper is concerned with a shape optimization problem governed by a non-smooth PDE, i.e., the nonlinearity in the state equation is not necessarily differentiable. We follow the functional variational approach of [40] where the set of…

In this paper we provide a complete characterization of the regularity properties of the solutions associated to the homogeneous Dirichlet problem \begin{equation*} \begin{cases} \displaystyle - \Delta_1 u= h(u)f & \text{in } \Omega, \\…

In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle…