arXiv subjects

Shuntaro Tsubouchi

Publications and source records attributed to Shuntaro Tsubouchi.

7 recordsLinked to original sources

arXiv2025

Gradient continuity for the parabolic $(1,\,p)$-Laplace system

This paper deals with the parabolic $(1,\,p)$-Laplace system, a parabolic system that involves the one-Laplace and $p$-Laplace operators with $p\in(1,\,\infty)$. We aim to prove that a spatial gradient is continuous in space and time. An external force term is treated under the optimal regularity assumption in the parabolic Lebesgue spaces. We also discuss a generalized parabolic system with the Uhlenbeck structure. A main difficulty is that the uniform ellipticity of the $(1,\,p)$-Laplace operator is violated on a facet, or the degenerate region of a spatial gradient. The gradient continuity is proved by showing local Hölder continuity of a truncated gradient, whose support is far from the facet. This is rigorously demonstrated by considering approximate parabolic systems and deducing various regularity estimates for approximate solutions by classical methods such as De Giorgi's truncation, Moser's iteration, and freezing coefficient arguments. A weak maximum principle is also utilized when $p$ is not in the supercritical range.

math.AP↗

arXiv2025

Continuity of the spatial gradient of weak solutions to very singular parabolic equations involving the one-Laplacian

We consider weak solutions to very singular parabolic equations involving a one-Laplace-type operator, which is singular and degenerate, and a $p$-Laplace-type operator with $\frac{2n}{n+2}<p<\infty$, where $n\ge 2$ denotes the space dimension. This type of equation is used to describe the motion of a Bingham flow. It has been a long-standing open problem of whether the spatial gradients of weak solutions are continuous in space and time. This paper aims to give an affirmative answer for a wide class of such equations. This equation becomes no longer uniformly parabolic near the facet, the place where the spatial gradient vanishes. To achieve our goal, we show local a priori Hölder continuity of gradients suitably truncated near facets. For this purpose, we consider a parabolic approximate problem and appeal to standard methods, including De Giorgi's truncation and comparisons with Dirichlet heat flows. Our method is a parabolic adjustment of our method developed to prove the corresponding statements for stationary problems.

math.AP↗

arXiv2024

Gradient continuity for the parabolic $(1,\,p)$-Laplace equation under the subcritical case

This paper is concerned with the gradient continuity for the parabolic $(1,\,p)$-Laplace equation. In the supercritical case $\frac{2n}{n+2} n(2-p)/p$. The gradient continuity is proved, similarly to the supercritical case, once the local gradient bounds of solutions are verified. Hence, this paper mainly aims to show the local boundedness of a solution and its gradient by Moser's iteration. The proof is completed by considering a parabolic approximate problem, and showing a priori gradient estimates of a bounded weak solution to the relaxed equation.

math.AP↗

arXiv2023

Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity

In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including one-Laplacian, and is perturbed by a $p$-Laplacian-type diffusion operator with $1<p<\infty$. This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use standard methods including De Giorgi's truncation and freezing coefficient methods.

math.AP↗

arXiv2022

A weak solution to a perturbed one-Laplace system by $p$-Laplacian is continuously differentiable

In this paper we aim to show continuous differentiability of weak solutions to a one-Laplace system perturbed by $p$-Laplacian with $1<p<\infty$. The main difficulty on this equation is that uniform ellipticity breaks near a facet, the place where a gradient vanishes. We would like to prove that derivatives of weak solutions are continuous even across the facets. This is possible by estimating Hölder continuity of Jacobian matrices multiplied with its modulus truncated near zero. To show this estimate, we consider an approximated system, and use standard methods including De Giorgi's truncation and freezing coefficient arguments.

math.AP↗

arXiv2022

Continuity of derivatives of a convex solution to a perturbed one-Laplace equation by $p$-Laplacian

We consider a one-Laplace equation perturbed by $p$-Laplacian with $1<p<\infty$. We prove that a weak solution is continuously differentiable ($C^{1}$) if it is convex. Note that similar result fails to hold for the unperturbed one-Laplace equation. The main difficulty is to show $C^{1}$-regularity of the solution at the boundary of a facet where the gradient of the solution vanishes. For this purpose we blow-up the solution and prove that its limit is a constant function by establishing a Liouville-type result, which is proved by showing a strong maximum principle. Our argument is rather elementary since we assume that the solution is convex. A few generalization is also discussed.

math.AP↗

arXiv2020

Local Lipschitz bounds for solutions to certain singular elliptic equations involving one-Laplacian

In this paper local Lipschitz regularity of weak solutions to certain singular elliptic equations involving one-Laplacian is studied. Equations treated here also contains another well-behaving elliptic operator such as $p$-Laplacian with $1<p<\infty$. The problem is that one-Laplacian is too singular on degenerate points, what is often called facet, which makes it difficult to obtain even Lipschitz regularity of weak solutions. This difficulty is overcome by making suitable approximation schemes, and by avoiding analysis on facet for approximated solutions. The key estimate is a local a priori uniform Lipschitz estimate for classical solutions to regularized equations, which is proved by Moser's iteration. Another local a priori uniform Lipschitz bounds can also be obtained by De Giorgi's truncation. Proofs of local Lipschitz estimates in this paper are rather classical and elementary in the sense that nonlinear potential estimates are not used at all.

math.AP↗