Related papers: An elliptic regularity theorem for fractional part…

This work is about global H\"older regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the…

We study Cauchy problems associated to elliptic operators acting on vector-valued functions and coupled up to the first-order. We prove pointwise estimates for the spatial derivatives of the semigroup associated to these problems in the…

We identify a large class of constant (complex) coefficient, second order elliptic systems for which the Dirichlet problem in the upper-half space with data in $L^p$-based Sobolev spaces, $1<p<\infty$, of arbitrary smoothness $\ell$, is…

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial…

In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the…

We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number $0< \varepsilon <1$, depending only…

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$ domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of…

Given a compact metric graph $\Gamma$ and the Laplacian $\Delta_{\Gamma}$ coupled with standard (Kirchhoff) vertex conditions, solutions to fractional elliptic partial differential equations of the form $(\kappa^2 -…

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…

We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of…

We investigate fractional regularity estimates up to the boundary for solutions to fully nonlinear elliptic equations with measurable ingredients. Specifically, under the assumption of uniform ellipticity of the operator, we demonstrate…

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…

We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their…

We establish existence, uniqueness, and Sobolev and H\"older regularity results for the stochastic partial differential equation $$ du=\left(\sum_{i,j=1}^d a^{ij}u_{x^ix^j}+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt+\sum_{k=1}^{\infty}g^kdw^k_t,…

We prove a sharp interior regularity theorem for fully nonlinear maximally subelliptic PDE in non-isotropic Sobolev spaces adapted to maximally subelliptic operators. This result complements the regularity theorem proven by Street for…

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is…

We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2}…

We present an $L_{p}$-theory ($p\geq 2$) for time-fractional stochastic partial differential equations driven by L\'evy processes of the type $$ \partial^{\alpha}_{t}u=\sum_{i,j=1}^d a^{ij}u_{x^{i}x^{j}}…

In this paper, we prove $L^p$ estimates for the fractional derivatives of solutions to elliptic fractional partial differential equations whose coefficients are $VMO$. In particular, our work extends the optimal regularity known in the…

We consider an elliptic equation with the fractional Laplacian operator $(-\Delta)^{\frac{\alpha}{2}}$ in the dissipative term, a singular integral operator ${\bf A}(\cdot)$ in the nonlinear term, and an external source $f$. The key example…