arXiv subjects
Publications and source records attributed to Sun-Sig Byun.
We prove the self-improving property of very weak solutions to non-uniformly elliptic problems of double phase type in divergence form under sharp assumptions on the nonlinearity.
We study $s$-fractional $p$-Laplacian type equations with discontinuous kernel coefficients in divergence form to establish $W^{s+σ,q}$ estimates for any choice of pairs $( σ,q)$ with $q\in(p,\infty)$ and $σ\in\left(0,\min\left\{\frac{s}{p-1},1-s\right\}\right)$ under the assumption that the associated kernel coefficients have small BMO seminorms near the diagonal. As a consequence, we find in the literature an optimal fractional Sobolev regularity of such a non-homogeneous nonlocal equation when the right-hand side is presented by a suitable fractional operator. Our results are new even in the linear case.
We study a class of nonlocal double phase problems with discontinuous coefficients. A local self-improving property and a higher Hölder continuity result for weak solutions to such problems are obtained under the assumptions that the associated coefficient functions are of type VMO (vanishing mean oscillation) and that the principal coefficient depends not only on the variables but also on the solution itself.
We investigate the De Giorgi-Nash-Moser theory for minimizers of mixed local and nonlocal functionals modeled after \[ v \mapsto \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\dfrac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dxdy+\int_Ωa(x)|Dv|^{q}\,dx, \] where $0<s<1<p \le q$ and $a(\cdot) \ge 0$. In particular, we prove Hölder regularity and Harnack's inequality under possibly sharp assumptions on $s,p,q$ and $a(\cdot)$.
We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, α}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator.
We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.
We establish a gradient estimate for a very weak solution to a quasilinear elliptic equation with a nonstandard growth condition, which is a natural generalization of the $p$-Laplace equation. We investigate the maximum extent for the gradient estimate to hold without imposing any regularity assumption on the nonlinearity other than basic structure assumptions. Our results also include a higher integrability result of the gradient and the existence for the very weak solutions to such nonlinear problems.
In this paper we are concerned with global maximal regularity estimates for elliptic equations with degenerate weights. We consider both the linear case and the non-linear case. We show that higher integrability of the gradients can be obtained by imposing a local small oscillation condition on the weight and a local small Lipschitz condition on the boundary of the domain. Our results are new in the linear and non-linear case. We show by example that the relation between the exponent of higher integrability and the smallness parameters is sharp even in the linear or the unweighted case.
We study generalized fractional $p$-Laplacian equations to prove local boundedness and Hölder continuity of weak solutions to such nonlocal problems by finding a suitable fractional Sobolev-Poincaré inquality.
We prove local boundedness and Hölder continuity for weak solutions to nonlocal double phase problems concerning the following fractional energy functional \[ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}} + a(x,y)\frac{|v(x)-v(y)|^q}{|x-y|^{n+tq}}\, dxdy, \] where $0<s\le t<1<p \leq q<\infty$ and $a(\cdot,\cdot) \geq 0$. For such regularity results, we identify sharp assumptions on the modulating coefficient $a(\cdot,\cdot)$ and the powers $s,t,p,q$ which are analogous to those for local double phase problems.
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.
We provide comprehensive regularity results and optimal conditions for a general class of functionals involving Orlicz multi-phase of the type \begin{align} \label{abst:1} v\mapsto \int_Ω F(x,v,Dv)\,dx, \end{align} exhibiting non-standard growth conditions and non-uniformly elliptic properties. The model functional under consideration is given by the Orlicz multi-phase integral \begin{align} \label{abst:2} v\mapsto \int_Ω f(x,v)\left[ G(|Dv|) + \sum\limits_{k=1}^{N}a_k(x)H_{k}(|Dv|) \right]\,dx,\quad N\geqslant 1, \end{align} where $G,H_{k}$ are $N$-functions and $ 0\leqslant a_{k}(\cdot)\in L^{\infty}(Ω) $ with $0 < ν\leqslant f(\cdot) \leqslant L$. Its ellipticity ratio varies according to the geometry of the level sets $\{a_{k}(x)=0\}$ of the modulating coefficient functions $a_{k}(\cdot)$ for every $k\in \{1,\ldots,N\}$. We give a unified treatment to show various regularity results for such multi-phase problems with the coefficient functions $\{a_{k}(\cdot)\}_{k=1}^{N}$ not necessarily Hölder continuous even for a lower level of the regularity. Moreover, assuming that minima of the functional above belong to better spaces such as $C^{0,γ}(Ω)$ or $L^κ(Ω)$ for some $γ\in (0,1)$ and $κ\in (1,\infty]$, we address optimal conditions on nonlinearity for each variant under which we build comprehensive regularity results. On the other hand, since there is a lack of homogeneity properties in the nonlinearity, we consider an appropriate scaling with keeping the structures of the problems under which we apply Harmonic type approximation in the setting varying on the a priori assumption on minima. We believe that the methods and proofs developed in this paper are suitable to build regularity theorems for a larger class of non-autonomous functionals.
We study quasilinear elliptic double obstacle problems with a variable exponent growth when the right-hand side is a measure. A global Calderón-Zygmund estimate for the gradient of an approximable solution is obtained in terms of the associated double obstacles and a given measure, identifying minimal requirements for the regularity estimate.
We study a generalized Stokes system with Orlicz growth which is nonstandard in a non-smooth domain. Our purpose is to derive a Calderon-Zygmund type estimate of the gradient of a solution and the pressure to such a system like (1.1) under a small BMO non-linearity and a sufficient flatness on the boundary of the domain. In the process, we overcome not only lack of Lipschitz regularity for the corresponding limiting problem, but also the fact that the associated structure depends only on the symmetric part of the gradient for the desired global estimate, which is new even in the literature of elliptic system on a nonsmooth domain.
We study fully nonlinear parabolic equations in nondivergence form with oblique boundary conditions. An optimal and global Calderón-Zygmund estimate is obtained by proving that the Hessian of the viscosity solution to the oblique boundary problem is as integrable as the nonhomogeneous term in $L^{p}$ spaces under minimal regularity requirement on the nonlinear operator, the boundary data and the boundary of the domain.
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.
This paper concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness and $W^{2,p}$-regularity results by finding approximate non-obstacle problems with the same oblique boundary condition and then making a suitable limiting process.
We obtain Calderón-Zygmund type estimates in generalized Morrey spaces for nonlinear equations of $p$-Laplacian type. Our result is obtained under minimal regularity assumptions both on the operator and on the domain. This result allows us to study asymptotically regular operators. As a byproduct, we obtain also generalized Hölder regularity of the solutions under some minimal restrictions of the weight functions.