arXiv subjects

Pilsoo Shin

Publications and source records attributed to Pilsoo Shin.

6 recordsLinked to original sources

arXiv2022

Regularity estimates for singular parabolic measure data problems with sharp growth

We prove global gradient estimates for parabolic $p$-Laplace type equations with measure data, whose model is $$u_t - \textrm{div} \left(|Du|^{p-2} Du\right) = μ\quad \textrm{in} \ Ω\times (0,T) \subset \mathbb{R}^n \times \mathbb{R},$$ where $μ$ is a signed Radon measure with finite total mass. We consider the singular case $$\frac{2n}{n+1} <p \le 2-\frac{1}{n+1}$$ and give possibly minimal conditions on the nonlinearity and the boundary of $Ω$, which guarantee the regularity results for such measure data problems.

math.AP↗

arXiv2022

Gradient type estimates for linear elliptic systems from composite materials

In this paper, we consider linear elliptic systems from composite materials where the coefficients depend on the shape and might have the discontinuity between the subregions. We derive a function which is related to the gradient of the weak solutions and which is not only locally piecewise Hölder continuous but locally Hölder continuous. The gradient of the weak solutions can be estimated by this derived function and we also prove local piecewise gradient Hölder continuity which was obtained by the previous results.

math.AP↗

arXiv2022

A geometric result for composite materials with $C^{1,γ}$-boundaries

In this paper, we obtain a geometric result for composite materials related to elliptic and parabolic partial differential equations. In the classical papers Li and Vogelius (2000), and Li and Nirenberg (2003), they assumed that for any scale and for any point there exists a coordinate system such that the boundaries of the individual components of a composite material locally become $C^{1,γ}$-graphs. We prove that if the individual components of a composite material are composed of $C^{1,γ}$-boundaries then such a coordinate system in Li and Vogelius (2000), and Li and Nirenberg (2003) exists, and therefore obtaining the gradient boundedness and the piecewise gradient Hölder continuity results for linear elliptic systems related to composite materials.

math.AP↗

arXiv2021

Global Hoelder continuity of solutions to quasilinear equations with Morrey data

We deal with general quasilinear divergence-form coercive operators whose prototype is the $m$-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property. We prove global boundedness and Hoelder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical $L^p$-result of Ladyzhenskaya and Ural'tseva to the settings of the Morrey spaces.

math.AP↗

arXiv2021

Global regularity for degenerate/singular parabolic equations involving measure data

We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity estimates for the spatial gradient of solutions to such parabolic measure data problems, by introducing the (intrinsic) fractional maximal function of a given measure.

math.AP↗

arXiv2017

Global Sobolev regularity for general elliptic equations of $p$-Laplacian type

We derive global gradient estimates for $W^{1,p}_0(Ω)$-weak solutions to quasilinear elliptic equations of the form $$ \mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over $n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to $x$ and Hölder continuous in $u.$ In the case when $p\geq n,$ we allow only continuous nonlinearity in $u.$ Our result highly improves the known regularity results available in the literature. In fact, we are able not only to weaken the regularity requirement on the nonlinearity in $u$ from Lipschitz continuity to Hölder one, but we also find a very lower level of geometric assumptions on the boundary of the domain to ensure global character of the obtained gradient estimates.

math.AP↗