Related papers: Maximal $L_p$-regularity for fractional problem dr…

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously,…

We establish an explicit maximum principle for the Dirichlet problem associated with the $p$-Laplacian ($p>1$), where the constant depends on both $p$ and the geometry of the domain. From this result we derive two main applications. First,…

By virtue of barrier arguments we prove $C^\alpha$-regularity up to the boundary for the weak solutions of a non-local nonlinear problem driven by the fractional $p$-Laplacian operator. The equation is boundedly inhomogeneous and the…

We establish local $C^{1,\alpha}$-regularity for some $\alpha\in(0,1)$ and $C^{\alpha}$-regularity for any $\alpha\in(0,1)$ of local minimizers of the functional \[ v\ \mapsto\ \int_\Omega \phi(x,|Dv|)\,dx, \] where $\phi$ satisfies a…

We prove $L^2$-maximal regularity of linear non-autonomous evolutionary Cauchy problem \begin{equation}\label{eq00}\nonumber \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator…

In this paper we consider an SPDE where the leading term is a second order operator with periodic boundary conditions, coefficients which are measurable in $(t,\omega)$, and H\"older continuous in space. Assuming stochastic parabolicity…

In this paper, we study the $\ell^p$-maximal regularity for the fractional difference equation with finite delay: \begin{equation*} \ \ \ \ \ \ \ \ \left\{\begin{array}{cc} \Delta^{\alpha}u(n)=Au(n)+\gamma u(n-\lambda)+f(n), \ n\in \mathbb…

We consider a class of nonlinear integro-differential equations whose leading operator is obtained as a superposition of $(-\Delta_{p})^{s}$ and $(-\Delta_{p})^{t}$, where $0<s<t<1<p<\infty$, weighted via two possibly degenerate…

We consider the Dirichlet problem Lu = 0 in D u = g on E = boundary of D for two second order elliptic operators L_k(u) = \sum_{i,j=1}^n a_k^{ij}(x) \partial_{ij} u(x), k=0,1, in a bounded Lipschitz domain D in R^n. The coefficients…

The aim of this paper is to exhibit a necessary and sufficient condition of optimality for functionals depending on fractional integrals and derivatives, on indefinite integrals and on presence of time delay. We exemplify with one example,…

We study an unbounded operator arising naturally after linearizing the system modelling the motion of a rigid body in a viscous incompressible fluid. We show that this operator is $\mathcal{R}$ sectorial in $L^q$ for every $q\in…

In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We…

We establish optimal L^p bounds for the nontangential maximal function of the gradient of the solution to a second order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the…

This paper studies a maximal $L^q$-regularity property for nonlinear elliptic equations of second order with a zero-th order term and gradient nonlinearities having superlinear and sub-quadratic growth, complemented with Dirichlet boundary…

We introduce and study the logarithmic $p$-Laplacian $L_{\Delta_p}$, which emerges from the formal derivative of the fractional $p$-Laplacian $(-\Delta_p)^s$ at $s=0$. This operator is nonlocal, has logarithmic order, and is the nonlinear…

We establish a partial $C^{1,\alpha}$ regularity result for minimizers of the optimal $p$-compliance problem with length penalization in any spatial dimension $N\geq 2$, extending some of the results obtained in…

Higher Sobolev and H\"older regularity is studied for local weak solutions of the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$. Depending on the regime considered, i.e. $$0<s\le\tfrac{p-2}{p}\quad \text{or}…

We study boundary regularity for the inhomogeneous Dirichlet problem for $2s$-stable operators in generalized H\"older spaces. Moreover, we provide explicit counterexamples that showcase the sharpness of our results. Our approach directly…

We introduce an $L_q(L_p)$-theory for the quasi-linear fractional equations of the type $$ \partial^{\alpha}_t u(t,x)=a^{ij}(t,x)u_{x^i x^j}(t,x)+f(t,x,u), \quad t>0, \,x\in \mathbf{R}^d. $$ Here, $\alpha\in (0,2)$, $p,q>1$, and…

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. Proofs…