arXiv subjects
Publications and source records attributed to Jihoon Ok.
We establish local regularity theory for parabolic systems of Uhlenbeck type with $φ$-growth. In particular, we prove local boundedness of weak solutions and their gradient, and then local Hölder continuity of the gradients, providing suitable assumptions on the growth function $φ$. Our approach, being independent of the degeneracy of the system, allows for a unified treatment of both the degenerate and the singular case.
We study weak solutions and minimizers $u$ of the non-autonomous problems $\operatorname{div} A(x, Du)=0$ and $\min_v \int_ΩF(x,Dv)\,dx$ with quasi-isotropic $(p, q)$-growth. We consider the case that $u$ is bounded, Hölder continuous or lies in a Lebesgue space and establish a sharp connection between assumptions on $A$ or $F$ and the corresponding norm of $u$. We prove a Sobolev--Poincaré inequality, higher integrability and the Hölder continuity of $u$ and $Du$. Our proofs are optimized and streamlined versions of earlier research that can more readily be further extended to other settings. Connections between assumptions on $A$ or $F$ and assumptions on $u$ are known for the double phase energy $F(x, ξ)=|ξ|^p + a(x)|ξ|^q$. We obtain slightly better results even in this special case. Furthermore, we also cover perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, triple phase, double variable exponent as well as variable exponent double phase energies and the results are new in most of these special cases.
We prove the partial Hölder continuity on boundary points for minimizers of quasiconvex non-degenerate functionals \begin{equation*} \mathcal{F}({\bf u}) \colon =\int_Ω f(x,{\bf u},D{\bf u})\,\mathrm{d}x, \end{equation*} where $f$ satisfies a uniform VMO condition with respect to the $x$-variable, is continuous with respect to ${\bf u}$ and has a general growth with respect to the gradient variable.
We establish maximal local regularity results of weak solutions or local minimizers of \[ \operatorname{div} A(x, Du)=0 \quad\text{and}\quad \min_u \int_ΩF(x,Du)\,dx, \] providing new ellipticity and continuity assumptions on $A$ or $F$ with general $(p,q)$-growth. Optimal regularity theory for the above non-autonomous problems is a long-standing issue; the classical approach by Giaquinta and Giusti involves assuming that the nonlinearity $F$ satisfies a structure condition. This means that the growth and ellipticity conditions depend on a given special function, such as $t^p$, $φ(t)$, $t^{p(x)}$, $t^p+a(x)t^q$, and not only $F$ but also the given function is assumed to satisfy suitable continuity conditions. Hence these regularity conditions depend on given special functions. In this paper we study the problem without recourse to special function structure and without assuming Uhlenbeck structure. We introduce a new ellipticity condition using $A$ or $F$ only, which entails that the function is quasi-isotropic, i.e.\ it may depend on the direction, but only up to a multiplicative constant. Moreover, we formulate the continuity condition on $A$ or $F$ without specific structure and without direct restriction on the ratio $\frac qp$ of the parameters from the $(p,q)$-growth condition. We establish local $C^{1,α}$-regularity for some $α\in(0,1)$ and $C^α$-regularity for any $α\in(0,1)$ of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases.
We derive weighted Sobolev-Poincaré type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class, so-called the parabolic $A_p$-class, where only the parabolic cubes are involved in the definition.
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 consider nonlinear elliptic equations of the $p$-Laplacian type with lower order terms which involve nonnegative potentials satisfying a reverse Hölder type condition. Then we obtain interior and boundary $L^q$ estimates for the gradient of weak solutions and the lower order terms, independently, under sharp regularity conditions on the coefficients and the boundaries. In particular, the proof in this paper does not employ Fefferman-Phong type inequalities which are essential tools in the linear cases in [3,47].
We obtain Calderón-Zygmund type estimates for parabolic equations with Orlicz growth, where nonlinearities involved in the equations may be discontinuous for the space and time variables. In addition, we consider parabolic systems with the Uhlenbeck structure.
We prove the local boundedness and the local Hölder continuity of weak solutions to nonlocal equations with variable orders and exponents under sharp assumptions.
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 establish local $C^{1,α}$-regularity for some $α\in(0,1)$ and $C^α$-regularity for any $α\in(0,1)$ of local minimizers of the functional \[ v\ \mapsto\ \int_Ωϕ(x,|Dv|)\,dx, \] where $ϕ$ satisfies a $(p,q)$-growth condition. Establishing such a regularity theory with sharp, general conditions has been an open problem since the 1980s. In contrast to previous results, we formulate the continuity requirement on $ϕ$ in terms of a single condition for the map $(x,t)\mapsto ϕ(x,t)$, rather than separately in the $x$- and $t$-directions. Thus we can obtain regularity results for functionals without assuming that the gap $\frac qp$ between the upper and lower growth bounds is close to $1$. Moreover, for $ϕ(x,t)$ with particular structure, including $p$-, Orlicz-, $p(x)$- and double phase-growth, our single condition implies known, essentially optimal, regularity conditions. Hence, we handle regularity theory for the above functional in a universal way.
We consider nonlinear elliptic equations that are naturally obtained from the elliptic Schrödinger equation $-Δu +Vu=0$ in the setting of the calculus of variations, and obtain $L^q$-estimates for the gradient of weak solutions. In particular, we generalize a result of Shen in [Ann. Inst. Fourier 45 (1995), no. 2, 513--546] in the nonlinear setting by using a different approach. This allows us to consider discontinuous coefficients with a small BMO semi-norm and non-smooth boundaries which might not be Lipschitz continuous.
We prove higher integrability of the spatial gradient of weak solutions to parabolic systems with $ϕ$-growth, where $φ=φ(t)$ is a general Orlicz function. The parabolic systems need be neither degenerate nor singular. Our result is a generalized version of the one of J. Kinnunen and J. Lewis [Duke Math. J. 102 (2000), no. 2, 253--271] for the parabolic $p$-Laplace systems.