Related papers: The Dirichlet problem for stochastic partial diffe…
We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the…
This paper deals with the spatial and temporal regularity of the unique Hilbert space valued mild solution to a semilinear stochastic partial differential equation with nonlinear terms that satisfy global Lipschitz conditions. It is shown…
We show global and interior higher-order log-H\"older regularity estimates for solutions of Dirichlet integral equations where the operator has a nonintegrable kernel with a singularity at the origin that is weaker than that of any…
Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric…
This paper deals with the \emph{integral} version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the H\"older regularity of the data. By…
We consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation invariant operators. For a bounded $C^2$ domain $\Omega \subset \mathbb{R}^d,$ let $u\in C(\mathbb{R}^d)$ be a viscosity solution of such Dirichlet…
In this paper we survey some results on the Dirichlet problem \[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }\Omega \\ u&=&g&\textrm{in }\mathbb R^n\backslash\Omega \end{array}\right.\] for nonlocal operators of the form…
We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains…
We settle the issue of well-posedness for the Dirichlet problem for a higher order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $\Omega$, with boundary data in Besov…
Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(\alpha)}_\kappa u+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \…
We study the Cauchy problem for a generalized derivative nonlinear Schr\"odinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions are constructed…
We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power $s>0$ of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders…
After recalling basic features of the theory of symmetric quasi regular Dirichlet forms we show how by applying it to the stochastic quantization equation, with Gaussian space-time noise, one obtains weak solutions in a large invariant set.…
The fractional Laplacian $(-\Delta )^a$, $a\in(0,1)$, and its generalizations to variable-coefficient $2a$-order pseudodifferential operators $P$, are studied in $L_q$-Sobolev spaces of Bessel-potential type $H^s_q$. For a bounded open set…
In this article, we study the regularity of solutions to inhomogeneous time-fractional evolution equations involving anisotropic non-local operators in mixed-norm Sobolev spaces of variable order, with non-trivial initial conditions. The…
A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that…
Using a standard linearization technique and previously obtained microlocal properties for pseudodifferential operators with smooth coefficients, the authors state results of microlocal regularity in generalized Besov spaces for solutions…
The asymptotic analysis of a class of stochastic partial differential equations (SPDEs) with fully locally monotone coefficients covering a large variety of physical systems, a wide class of quasilinear SPDEs and a good number of fluid…
In this article, we present a simpler and alternative proof of the solvability of the regularity problem - that is, the Dirichlet problem with boundary data in $\dot W^{1,p}$ - for uniformly elliptic operators on $\mathbb{R}^n_+$ under a…
A priori estimates for the weak solutions the Dirichlet problem for the uniformly higher-order elliptic equations in a smooth bounded domain $\Omega\subset \Rn$ in generalized weighted Sobolev-Morrey spaces are obtained.