arXiv subjects
Publications and source records attributed to Martin Ulmer.
We use convex integration techniques to provide examples of failure of weighted Calder\'{o}n-Zygmund estimates for degenerate linear elliptic PDEs when the weights are in $A_p$, $p > 2$.
We consider an elliptic operator $L$ with variable, merely bounded, and measurable coefficients on a Lipschitz domain, and study solutions to $Lu=0$ that attain given Neumann and Dirichlet-regularity data on different parts of the boundary. The boundary data lies in $L^p$ or $W^{1,p}$ respectively, and we show nontangential maximal function estimates of the gradient of the solution. This mixed boundary value problem generalizes the pure Dirichlet, regularity, and Neumann problem with rough boundary data in $L^p$, and the already established mixed boundary value problem for the Laplacian.
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, Shen (2007) established a partial reverse result. In our work, we show that if we assume an $L^1$-Carleson condition on only $|\partial_t A|$ the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D)^*_{p'}$ and the Regularity problem $(R)_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^\infty$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this $L^1$-Carleson condition on $|\partial_t A|$.
In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $\partial_tu - \mbox{div}(A\nabla u)=0$ on the domain $\mathbb R^{n+1}_+\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients are independent of the spatial variable $x_{n+1}$ (which is transversal to the boundary). We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. An analogous result for the Dirichlet problem has been considered earlier by Auscher, Egert and Nystr\"om, however the Regularity problem represents an additional step up in difficulty. In the elliptic case, the analog of the question considered here was resolved for both Dirichlet and Regularity problems by Hofmann, Kenig, Mayboroda and Pipher. The main result of this paper complements a recent work of two of the authors with L. Li showing solvability of the parabolic Regularity problem for data in some $L^p$ spaces when the coefficients satisfy a natural Carleson condition (which is a parabolic analog of the so-called DKP-condition).
We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that if the matrix $A$ is independent in the transversal $t$-direction, then the regularity boundary value problem is solvable with data in a Sobolev space. In the present paper we improve on the $t$-independence condition by introducing a mixed $L^1-L^\infty$ condition that only depends on $\partial_t A$, the derivative of $A$ in transversal direction. This condition is different from other conditions in the literature and has already been proven to imply solvability of the Dirichlet boundary value problem.
We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half plane $\mathbb{R}^2_+$. There are several conditions on the behavior of the matrix $A$ in the transversal $t$-direction that yield $\omega\in A_\infty(\sigma)$. These include the $t$-independence condition, a mixed $L^1-L^\infty$ condition on $\partial_t A$, and Dini-type conditions. We introduce an $L^1$ Carleson condition on $\partial_t A(x,t)$ that extends the class of elliptic operators for which we have $\omega\in A_\infty(\sigma)$, i.e. solvability of the $L^p$ Dirichlet problem for some $1<p<\infty$.
We show small and large Carleson perturbation results for the parabolic Regularity boundary value problem with boundary data in $\dot{L}_{1,1/2}^p$ and small Carelson perturbation results for the Neumann problem with boundary data in $L^p$. The operator we consider is $L:=\partial_t -\mathrm{div}(A\nabla\cdot)$ and the domains are parabolic cylinders $\Omega=\mathcal{O}\times\mathbb{R}$, where $\mathcal{O}$ is a Lipschitz domain.
We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that if the matrix $A$ is independent in the transversal $t$-direction, then we have $\omega\in A_\infty(\sigma)$. In the present paper we improve on the $t$-independence condition by introducing a mixed $L^1-L^\infty$ Carleson type condition that only depends on $\partial_t A$ and show $\omega\in A_\infty(\sigma)$ under this condition. This condition is different from other conditions in the literature. In the case of the upper half plane, we obtain the improvement that an $L^1$-Carleson condition on $|\partial_tA|$ implies $\omega\in A_\infty(\sigma)$. In particular, this condition is similar to an $L^1$-version of the DKP condition with derivative in only the transversal direction.
In the present paper we study perturbation theory for the $L^p$ Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators $L_0 = \mbox{div}(A_0\nabla)$ and $L_1 = \mbox{div}(A_1\nabla)$ such that the $L^p$ Dirichlet problem for $L_0$ is solvable for some $p>1$; we show that if $A_0 - A_1$ satisfies certain Carleson condition, then the $ L^q$ Dirichlet problem for $L_1$ is solvable for some $q \geq p$. Moreover if the Carleson norm is small then we may take $q=p$. We use the approach first introduced in Fefferman-Kenig-Pipher '91 on the unit ball, and build on Milakis-Pipher-Toro '11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the $L^p$ Dirichlet problem on a bounded Lipschitz domain for an operator $L = \mbox{div}(A\nabla)$, where $A$ satisfies a Carleson condition similar to the one assumed in Kenig-Pipher '01 and Dindo\v{s}-Petermichl-Pipher '07 but with unbounded antisymmetric part.