Sharp Sobolev Regularity for Widely Degenerate Parabolic Equations: A Study with Data in Lebesgue-Besov Spaces
This research paper establishes the Sobolev spatial regularity of weak solutions for a class of widely degenerate parabolic equations, including the evolutionary p-Poisson equation, under weaker assumptions on the data (belonging to Lebesgue-Besov spaces) than previously explored.
On Local Well-posedness of the Periodic Korteweg-de Vries Equation Below $H^{-\frac{1}{2}}(\mathbb{T})$: A Modulation-Restricted Normal Form Approach
This research paper presents a novel method for proving the local well-posedness of the periodic Korteweg-de Vries (KdV) equation for a specific range of low regularity initial data, contributing to the understanding of the equation's solutions in challenging scenarios.
Estimates Controlling a Function by its Radial Derivative: Applications to Stable Solutions of Elliptic Equations
This article presents new estimates that control a function (specifically, stable solutions of semilinear elliptic equations) using only its radial derivative, leading to a simplified and quantitative proof of Hölder regularity for these solutions.
Existence and Uniqueness of Weak Solutions for a Degenerate Goursat Type Linear Problem in Admissible Tricomi Domains
This research paper establishes the existence and uniqueness of weak solutions for a linear partial differential equation involving a degenerate Gellerstedt operator with mixed-type Dirichlet boundary conditions in a specific type of Tricomi domain.
Wild Solutions of Locally Infinite Energy to Scalar Euler-Lagrange Equations: Existence and Non-Existence Results
This paper investigates the existence of "wild" solutions, possessing locally infinite energy, to scalar Euler-Lagrange equations, establishing an optimal condition for their non-existence and demonstrating their existence outside this condition using convex integration.
Weighted Lp-Solvability of the Heat and Time-Fractional Heat Equations in Non-Smooth Domains: A Unified Approach Using Superharmonic Functions
This paper presents a unified framework for establishing the weighted Lp-solvability of the heat and time-fractional heat equations in a broad class of non-smooth domains characterized by the Hardy inequality. The key innovation lies in employing superharmonic Harnack functions to capture the boundary behavior of solutions, leading to weighted Lp estimates and solvability results applicable to various non-smooth domains, including those with exterior cone conditions, convex domains, and domains with specific capacity density properties.
On the L1-Stability of Parabolic Equations with Supercritical Drift Terms: Existence, Uniqueness, and Stability Analysis
This paper investigates the existence, uniqueness, and stability of weak solutions for parabolic equations with supercritical drift terms, highlighting the crucial role of the "non-spectral" condition (div b ≤ 0) in ensuring these properties.
Gradient and Laplacian Estimates for Singular Complex Monge-Ampère Equations
This paper presents a novel method for obtaining gradient and Laplacian estimates for solutions to singular complex Monge-Ampère equations using an integral approach and a Sobolev inequality with respect to the unknown metric.
A Rigidity Theorem for Kolmogorov-Type Operators: Proving the Uniqueness of L-balls
This research paper proves that a specific integral identity, involving the fundamental solution and a kernel function, uniquely characterizes a class of geometric shapes called L-balls for a family of hypoelliptic operators known as Kolmogorov-type operators.
Asymptotic Behavior of Solutions to Fully Nonlinear Elliptic PDEs in Thin Domains with Oblique Boundary Conditions
This research paper investigates the behavior of solutions to fully nonlinear elliptic partial differential equations (PDEs) in thin domains with oblique boundary conditions as the domain's thickness approaches zero. The authors derive the limit equation governing the behavior of solutions in the vanishing thickness limit and prove the convergence of solutions to the solution of the limit equation.
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