Related papers: A non-local approach to the generalized Stokes ope…
We prove existence of a solution to the divergence equation satisfying a new Bogovski-type estimate for the difference quotients. This enables us to give an alternative proof of the interior regularity of the solution to the $p$-Stokes…
We study Caccioppoli's inequalities of the non-stationary Stokes equations and Navier-Stokes equations. Our analysis is local near boundary and we prove that, in contrast to the interior case, the Caccioppoli's inequalities of the Stokes…
We study the $L^p$ mapping properties of the strong spherical maximal function, which is a multiparameter generalisation of Stein's spherical maximal function. We show that this operator is bounded on $L^p$ for $p > 2$ in all dimensions $n…
In this paper we establish $L^p$ boundedness properties for maximal operators, Littlewood-Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein-Uhlenbeck…
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 consider the steady Stokes equations supplemented with Navier boundary conditions including a non-negative friction coefficient. We prove maximal regularity estimates (including the prominent spaces $W^{1,p}$ and $W^{2,p}$ for…
In this paper we study the boundedness of global pseudo-differential operators on smooth manifolds. By using the notion of global symbol we extend a classical condition of H\"ormander type to guarantee the $L^p$-$L^q$-boundedness of global…
In this paper we provide an extension of a theorem of David and Semmes ('91) to general non-atomic measures. The result provides a geometric characterization of the non-atomic measures for which a certain class of square function operators,…
We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet…
In this work, we obtain quantitative estimates of the continuity constant for the $L^p$ maximal regularity of relatively continuous nonautonomous operators $\mathbb{A} : I \longrightarrow \mathcal{L}(D,X)$, where $D \subset X$ densely and…
We study a general class of elliptic free boundary problems equipped with a Dirichlet boundary condition. Our primary result establishes an optimal $C^{1,1}$-regularity estimate for $L^p$-strong solutions at points where the free and fixed…
A sequential quadratic programming (SQP) algorithm is designed for nonsmooth optimization problems with upper-C^2 objective functions. Upper-C^2 functions are locally equivalent to difference-of-convex (DC) functions with smooth convex…
Local behaviors near boundary are analyzed for solutions of the Stokes and Navier-Stoke equations in the half space with localized non-smooth boundary data. We construct solutions of Stokes equations whose velocity field is not bounded near…
We deal with a wide class of generalized nonlocal $p$-Laplace equations, so-called nonlocal $G$-Laplace equations, in the Heisenberg framework. Under natural hypotheses on the $N$-function $G$, we provide a unified approach to investigate…
We study $L^p$-theory of second-order elliptic divergence type operators with complex measurable coefficients. The major aspect is that we allow complex coefficients in the main part of the operator, too. We investigate generation of…
Ergodicity for local and nonlocal stochastic singular $p$-Laplace equations is proven, without restriction on the spatial dimension and for all $p\in[1,2)$. This generalizes previous results from [Gess, T\"{o}lle; J. Math. Pures Appl.,…
We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only…
The Fredholm index of unbounded operators defined on generalized solutions of nonlocal elliptic problems in plane bounded domains is investigated. It is known that nonlocal terms with smooth coefficients having zero of a certain order at…
We prove $L^p_{comp}\to L^p_{s}$ boundedness for averaging operators associated to a class of curves in the Heisenberg group $\mathbb{H}^1$ via $L^2$ estimates for related oscillatory integrals and Bourgain-Demeter decoupling inequalities…
We study Liouville theorems for the non-stationary Stokes equations in exterior domains in $ \mathbb{R}^{n}$ under decay conditions for spatial variables. As applications, we prove that the Stokes semigroup is a bounded analytic semigroup…