arXiv subjects

Georgios Sakellaris

Publications and source records attributed to Georgios Sakellaris.

8 recordsLinked to original sources

arXiv2025

Two-sided Gaussian estimates for fundamental solutions of second-order parabolic equations in non-divergence form

We establish two-sided Gaussian bounds for the fundamental solution of second-order parabolic operators in non-divergence form under minimal regularity assumptions. Specifically, we show that the upper and lower bounds follow from the local boundedness property and the weak Harnack inequality for the adjoint operator $P^*$, respectively. This provides a simpler and more direct proof of the Gaussian estimates when the coefficients have Dini mean oscillation in $x$, avoiding the use of normalized adjoint solutions required in previous works.

math.AP↗

arXiv2023

The Neumann Green function and scale invariant regularity estimates for elliptic equations with Neumann data in Lipschitz domains

We construct the Neumann Green function and establish scale invariant regularity estimates for solutions to the Neumann problem for the elliptic operator $Lu=-{\rm div}({\bf A} \nabla u+ \boldsymbol{b}u)+ \boldsymbol{c} \cdot \nabla u+du$ in a Lipschitz domain $Ω$. We assume that ${\bf A}$ is elliptic and bounded, that the lower order coefficients belong to scale invariant Lebesgue spaces, and that either $d\geq{\rm div}\boldsymbol{b}$ in $Ω$ and $\boldsymbol{b}\cdotν\geq 0$ on $\partialΩ$ in the sense of distributions, or the analogous condition for $\boldsymbol{c}$ holds. We develop the $L^2$ theory, construct the Neumann Green function and show estimates in the respective optimal spaces, and show local and global pointwise estimates for solutions. The main novelty is that our estimates are scale invariant, since our constants depend on the lower order coefficients only via their norms, and on the Lipschitz domain only via its Lipschitz character. Moreover, our pointwise estimates are shown in the optimal scale invariant setting for the inhomogeneous terms and the Neumann data.

math.AP↗

arXiv2020

Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces

We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation $-{\rm div}(A\nabla u+bu)+c\nabla u+du=-{\rm div}f+g$, assuming that $A$ is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions $b,f\in L^{n,1}$, $d,g\in L^{\frac{n}{2},1}$ and $c\in L^{n,q}$ for $q\leq\infty$, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with "good" constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary smallness condition on $b,d$ or $c,d$, we show a maximum principle and Moser's estimate for subsolutions with "good" constants. We also show the reverse Moser estimate for nonnegative supersolutions with "good" constants, under no smallness assumptions when $q<\infty$, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions. Finally, we show that, in the setting of Lorentz spaces, our assumptions are the sharp ones to guarantee these estimates.

math.AP↗

arXiv2019

Minimizers for the thin one-phase free boundary problem

We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb R^n$. We establish full regularity of the free boundary for dimensions $n \leq 2$, prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced in \cite{AltCaffarelli}. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE.

math.AP↗

arXiv2019

On scale invariant bounds for Green's function for second order elliptic equations with lower order coefficients and applications

We construct Green's functions for elliptic operators of the form $\mathcal{L}u=-\text{div}(A\nabla u+bu)+c\nabla u+du$ in domains $Ω\subseteq\mathbb R^n$, under the assumption $d\geq\text{div}b$, or $d\geq\text{div}c$. We show that, in the setting of Lorentz spaces, the assumption $b-c\in L^{n,1}(Ω)$ is both necessary and optimal to obtain pointwise bounds for Green's functions. We also show weak type bounds for Green's functions and their gradients. Our estimates are scale invariant and hold for general domains $Ω\subseteq\mathbb R^n$. Moreover, there is no smallness assumption on the norms of the lower order coefficients. As applications we obtain scale invariant global and local boundedness estimates for subsolutions to $\mathcal{L}u\leq -\text{div}f+g$ in the case $d\geq\text{div}c$.

math.AP↗

arXiv2018

Green's function for second order elliptic equations with singular lower order coefficients

We construct Green's function for second order elliptic operators of the form $Lu=-\nabla \cdot (\mathbf{A} \nabla u + \boldsymbol{b} u)+ \boldsymbol c \cdot \nabla u+ du$ in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients $\mathbf A$ is uniformly elliptic and bounded and the lower order coefficients $\boldsymbol{b}$, $\boldsymbol{c}$, and $d$ belong to certain Lebesgue classes and satisfy the condition $d - \nabla \cdot \boldsymbol{b} \ge 0$. In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green's function in the case when the mean oscillations of the coefficients $\mathbf A$ and $\boldsymbol{b}$ satisfy the Dini conditions and the domain is $C^{1, \rm{Dini}}$.

math.AP↗

arXiv2018

Boundary value problems in Lipschitz domains for equations with lower order coefficients

We use the method of layer potentials to study the $R_2$ Regularity problem and the $D_2$ Dirichlet problem for second order elliptic equations of the form $\mathcal{L}u=0$, with lower order coefficients, in bounded Lipschitz domains. For $R_2$ we establish existence and uniqueness assuming that $\mathcal{L}$ is of the form $\mathcal{L}u=-\text{div}(A\nabla u+bu)+c\nabla u+du$, where the matrix $A$ is uniformly elliptic and Hölder continuous, $b$ is Hölder continuous, and $c,d$ belong to Lebesgue classes and they satisfy either the condition $d\geq\text{div}b$, or $d\geq\text{div}c$ in the sense of distributions. In particular, $A$ is not assumed to be symmetric, and there is no smallness assumption on the norms of the lower order coefficients. We also show existence and uniqueness for $D_2$ for the adjoint equations $\mathcal{L}^tu=0$.

math.AP↗

arXiv2017

Boundary value problems in Lipschitz domains for equations with drifts

In this work we establish solvability and uniqueness for the $D_2$ Dirichlet problem and the $R_2$ Regularity problem for second order elliptic operators $L=-{\rm div}(A\nabla\cdot)+b\nabla\cdot$ in bounded Lipschitz domains, where $b$ is bounded, as well as their adjoint operators $L^t=-{\rm div}(A^t\nabla\cdot)-{\rm div}(b\,\cdot)$. The methods that we use are estimates on harmonic measure, and the method of layer potentials. The nature of our techniques applied to $D_2$ for $L$ and $R_2$ for $L^t$ leads us to impose a specific size condition on ${\rm div}b$ in order to obtain solvability. On the other hand, we show that $R_2$ for $L$ and $D_2$ for $L^t$ are uniquely solvable, assuming only that $A$ is Lipschitz continuous (and not necessarily symmetric) and $b$ is bounded.

math.AP↗