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Martin Dindoš

Publications and source records attributed to Martin Dindoš.

27 records · Page 2Linked to original sources
arXiv2018

Parabolic $L^p$ Dirichlet Boundary Value Problem and VMO-type time-varying domains

We prove the solvability of the parabolic $L^p$ Dirichlet boundary value problem for $1 < p \leq \infty$ for a PDE of the form $u_t = \mbox{div} (A \nabla u) + B \cdot \nabla u$ on time-varying domains where the coefficients $A= [a_{ij}(X, t)]$ and $B=[b_i]$ satisfy a certain natural small Carleson condition. This result brings the state of affairs in the parabolic setting up to the elliptic standard. Furthermore, we establish that if the coefficients of the operator $A,\,B$ satisfy a vanishing Carleson condition and the time-varying domain is of VMO type then the parabolic $L^p$ Dirichlet boundary value problem is solvable for all $1 < p \leq \infty$. This result is related to results in papers by Mazýa, Mitrea and Shaposhnikova, and Hofmann, Mitrea and Taylor where the fact that boundary of domain has normal in VMO or near VMO implies invertibility of certain boundary operators in $L^p$ for all $1 < p \leq \infty$ which then (using the method of layer potentials) implies solvability of the $L^p$ boundary value problem in the same range for certain elliptic PDEs. Our result does not use the method of layer potentials, since the coefficients we consider are too rough to use this technique but remarkably we recover $L^p$ solvability in the full range of $p$'s as the two papers mentioned above.

math.AP↗
arXiv2018

Regularity theory for solutions to second order elliptic operators with complex coefficients and the $L^p$ Dirichlet problem

We establish a new theory of regularity for elliptic complex valued second order equations of the form $\mathcal L=$div$A(\nabla\cdot)$, when the coefficients of the matrix $A$ satisfy a natural algebraic condition, a strengthened version of a condition known in the literature as $L^p$-dissipativity. Precisely, the regularity result is a reverse Hölder condition for $L^p$ averages of solutions on interior balls, and serves as a replacement for the De Giorgi - Nash - Moser regularity of solutions to real-valued divergence form elliptic operators. In a series of papers, Cialdea and Maz'ya studied necessary and sufficient conditions for $L^p$-dissipativity of second order complex coefficient operators and systems. Recently, Carbonaro and Dragičević introduced a condition they termed $p$-ellipticity, and showed that it had implications for boundedness of certain bilinear operators that arise from complex valued second order differential operators. Their $p$-ellipticity condition is exactly our strengthened version of $L^p$-dissipativity. The regularity results of the present paper are applied to solve $L^p$ Dirichlet problems for $\mathcal L=$div$A(\nabla\cdot)+B\cdot\nabla$ when $A$ and $B$ satisfy a natural and familiar Carleson measure condition. We show solvability of the $L^p$ Dirichlet boundary value problem for $p$ in the range where $A$ is $p$-elliptic.

math.AP↗
arXiv2017

Parabolic Regularity and Dirichlet boundary value problems

We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form $Lu=\text{div}(A \nabla u)-u_t=0$ in Lip$(1,1/2)$ time-varying cylinders, where the coefficient matrix $A = \left[ a_{ij}(X,t)\right] $ is uniformly elliptic and bounded. We show that if the Regularity problem $(R)_p$ for the equation $Lu=0$ is solvable for some $1<p<\infty$ then the Dirichlet problem $(D^*)_{p'}$ for the adjoint equation $L^*v=0$ is also solvable, where $p'=p/(p-1)$. This result is an analogue of the result established in the elliptic case by Kenig and Pipher. In the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by Hofmann and Lewis and Nyström for scalar parabolic systems. In comparison, our result is abstract with no assumption on the coefficients beyond the ellipticity condition and is valid in more general class of domains.

math.AP↗
arXiv2015

The Boundary value problems for second order elliptic operators satisfying a Carleson condition

Let $Ω$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\mbox{div} (A\nabla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in $H^{1,p}(\partialΩ)$ and of the Neumann problem with $L^p(\partialΩ)$ data for the operator $L$ on Lipschitz domains with small Lipschitz constant. We allow the coefficients of the operator $L$ to be rough obeying a certain Carleson condition with small norm. These results complete the results of [5] where $L^p(\partialΩ)$ Dirichlet problem was considered under the same assumptions and [6] where the regularity and Neumann problems were considered on two dimensional domains.

math.AP↗
arXiv2015

BMO solvability and the $A_\infty$ condition for second order parabolic operators

We prove that the $A_\infty$ property of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. Kircheim, Kenig, Pipher, and T. Toro established this criterion on bounded solutions in the elliptic case, improving an earlier result of Dindos, Kenig and Pipher for solutions with data in BMO. The extension to the parabolic setting requires an approach to the key estimate that primarily exploits the maximum principle. For various classes of parabolic operators, this criterion makes it easier to establish the solvability of the Dirichlet problem with data in $L^p$ for some $p$ (see results of Rivera-Noriega), and also to quantify these results in several aspects.

math.AP↗
arXiv2011

The regularity problem for elliptic operators with boundary data in Hardy-Sobolev space $HS^1$

Let $Ω$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in Hardy-Sobolev space $\HS$ is equivalent to the solvability of the Dirichlet regularity problem for boundary data in $H^{1,p}$ for some $1<p<\infty$. This is a "dual result" to a theorem in \cite{DKP09}, where it has been shown that the solvability of the Dirichlet problem with boundary data in $\text{BMO}$ is equivalent to the solvability for boundary data in $L^p(\partialΩ)$ for some $1<p<\infty$.

math.AP↗
arXiv2007

The sharp $A_p$ constant for weights in a reverse-Hölder class

In a recent paper V. Vasyunin presented a proof of the reverse Hölder inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function technique to find the sharp $A_p$ constants for weights in a reverse-Hölder class on an interval; we also find the sharp constants for the higher-integrability result of Gehring. Additionally, we find bounds for the $A_p$ constants of reverse-Hölder-class weights defined on rectangles and on cubes in n dimensions.

math.CA↗