Related papers: An $L_{q}(L_{p})$-regularity theory for parabolic …

In this paper we prove higher regularity for 2m-th order parabolic equations with general boundary conditions. This is a kind of maximal L_p-L_q regularity with differentiability, i.e. the main theorem is isomorphism between the solution…

We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $u_t = \I u$, where $\I$ is translation invariant and elliptic with respect to…

In this work, we investigate the qualitative properties as uniqueness, regularity and stabilization of the weak solution to the nonlinear parabolic problem involving general $p(x)$-homogeneous operators: \begin{equation*} \left\{…

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial-boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem…

In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form…

In this paper, we consider the regularity theory for fully nonlinear parabolic integro-differential equations with symmetric kernels. We are able to find parabolic versions of Alexandrov-Backelman-Pucci estimate with 0<\sigma<2. And we show…

In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $\Omega=\mathcal O\times\mathbb R$…

We consider non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$, where $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function. By using a purely analytic method, we prove the continuity of the non-local…

We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$ \partial_tu+\textrm{div}\big(f(u)\big)=\mathcal{L}[b(u)] $$ on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion…

The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular SPDEs of the form $$\partial_t u = \mathfrak{L} u+ F(u, \xi)\ ,$$ where the differential operator…

Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\Omega$. We consider the fractional nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~\Omega,\\…

We study the relationship between the Dirichlet and Regularity problem for parabolic operators of the form $ L = \mbox{div}(A\nabla\cdot) - \partial_t $ on cylindrical domains $ \Omega = \mathcal O \times \mathbb R $, where the base $…

In this manuscript, we establish the existence and sharp geometric regularity estimates for bounded solutions of a class of quasilinear parabolic equations in non-divergence form with non-homogeneous degeneracy. The model equation in this…

In this paper, we establish several Liouville-type theorems for a class of nonhomogenenous quasilinear inequalities. In the first part, we prove various Liouville results associated with nonnegative solutions to \begin{equation*}\tag{$P_s$}…

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order $2s$, with $s\in(0,1)$. We consider the class of nonlocal operators $\mathcal L_*\subset \mathcal L_0$, which…

We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-\Delta _{p(x)}u = f(x,u)$ in $ (0,T)\times\Omega$; $u = 0$ on $(0,T)\times\partial\Omega$; $u(0,x)=u_0(x)$ in $\Omega$;…

This paper is devoted to the study of $L^p$-maximal regularity for non-autonomous linear evolution equations of the form \begin{equation*}\label{Multi-pert1-diss-non} \dot u(t)+A(t)B(t)u(t)=f(t)\ \ t\in[0,T],\ \ u(0)=u_0. \end{equation*}…

In this article, we obtain higher H\"older regularity results for weak solutions to nonlocal problems driven by the fractional double phase operator \begin{align*} \mc L u(x):=&2 \; {\rm P.V.} \int_{\mathbb R^N}…

In this paper, we prove the unique existence and investigate the $L^{p}$-regularity of solutions to stochastic partial differential equations in Hilbert spaces associated with pseudo-differential operators, driven by Hilbert space-valued…

We prove that local weak solutions to nonlocal parabolic $p$-Laplace equations are locally Lipschitz continuous in space, uniformly in time for every $1<p<\infty$ and $s \in (0,1)$ whenever $sp > p-1$. Our results hold for symmetric,…