Search

We prove the Wolff potential estimates for nonlocal equations with nonstandard growth. As an application, we obtain the Wiener criterion in this framework, which provides a necessary and sufficient condition for boundary points to be regular. Our approach relies on the fine analysis of superharmonic functions in view of nonlocal nonlinear potential theory., Comment: 38 pages

We consider a flow by powers of Gauss curvature under the obstruction that the flow cannot penetrate a prescribed region, so called an obstacle. For all dimensions and positive powers, we prove the optimal curvature bounds of solutions and all time existence with its long time behavior. We also prove the $C^1$ regularity of free boundaries under a uniform thickness assumption., Comment: To appear in Math. Ann

In this paper, we investigate energy-minimizing curves with fixed endpoints $p$ and $q$ in a constrained space. We prove that when one of the endpoints, say $p$, is fixed, the set of points $q$ for which the energy-minimizing curve is not unique has no interior points.

In this paper, we establish the boundary regularity results for viscosity solutions of fully nonlinear degenerate/singular parabolic equations of the form $$u_t - x_n^{\gamma} F(D^2 u,x,t) = f,$$ where $\gamma<1$. These equations are motivated by the porous media type equations. We show the boundary $C^{1,\alpha}$-regularity of functions in their solutions class and the boundary $C^{2,\alpha}$-regularity of solutions. As an application, we derive the global regularity results and the solvability of the Cauchy-Dirichlet problems., Comment: 30 pages

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form $$u_t = a^{i'j'}u_{i'j'} + 2 x_n^{\gamma/2} a^{i'n} u_{i'n} + x_n^{\gamma} a^{nn} u_{nn} + b^{i'} u_{i'} + x_n^{\gamma/2} b^n u_{n} + c u + f \quad (\gamma \leq1).$$ When the equation above is singular, it can be derived from Monge--Amp\`ere equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution $u$, which is called $s$-polynomial. To prove $C_s^{2+\alpha}$-regularity and higher regularity of the solution $u$, we establish generalized Schauder theory which approximates coefficients of the operator with $s$-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap method to obtain higher regularity., Comment: 42 pages

We establish the interior $C^{1,\alpha}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^{\gamma}F(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized Dirichlet problem \begin{equation*} \left\{ \begin{aligned} u_t&=(1+|Du|^2)^{\gamma/2}F(D^2u) &&\text{in $Q_1$} \newline u&=\varphi &&\text{on $\partial_p Q_1$}. \end{aligned}\right. \end{equation*} Our approach utilizes the Bernstein method with approximations in view of difference quotient., Comment: 30 pages

We study the boundary continuity of solutions to fully nonlinear elliptic equations. We first define a capacity for operators in non-divergence form and derive several capacitary estimates. Secondly, we formulate the Wiener criterion, which characterizes a regular boundary point via potential theory. Our approach utilizes the asymptotic behavior of homogeneous solutions, together with Harnack inequality and the comparison principle., Comment: 35 pages

We present three equivalent definitions of the fractional $p$-Laplacian $(-\Delta_{\mathbb{H}^{n}})^{s}_{p}$, $01$, with normalizing constants, on the hyperbolic spaces. The explicit values of the constants enable us to study the convergence of the fractional $p$-Laplacian to the $p$-Laplacian as $s \to 1^{-}$., Comment: 23 pages

We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, \alpha}$-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., Comment: 23 pages

We establish an optimal C^{1,\alpha}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.

We study the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability $s \in (0, 1)$ and summability $p>1$. We establish a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory., Comment: 39 pages

We prove sharp near-diagonal pointwise bounds for the Green function $G_\Omega(x,y)$ for nonlocal operators of fractional order $\alpha \in (0,2)$. The novelty of our results is two-fold: the estimates are robust as $\alpha \to 2-$ and we prove the bounds without making use of the Dirichlet heat kernel $p_\Omega(t;x,y)$. In this way we can cover cases, in which the Green function satisfies isotropic bounds but the heat kernel does not., Comment: To appear in Int. Math. Res. Not. IMRN

The Krylov--Safonov theory for fully nonlinear nonlocal operators on hyperbolic spaces of dimension three is established. Since the operators on hyperbolic spaces exhibit qualitatively different behavior than those on manifolds with nonnegative curvature, new scale functions are introduced which take the effect of negative curvature into account. The regularity theory in this work provides unified regularity results for fractional-order and second-order operators in the sense that the regularity estimates stay uniform as the fractional-order approaches 2. In the unified regularity theory, the asymptotic behavior of the normalizing constant for the fractional Laplacian plays a fundamental role. The dimension restriction has been imposed to compute the explicit value of this constant by using the Fourier analysis on hyperbolic spaces., Comment: 44 pages

In this paper, we study the mathematical properties of the solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ to the degenerate parabolic system \begin{equation*} \bold{u}_t=\nabla\cdot\left(\left|\nabla\bold{u}\right|^{p-2}\nabla \bold{u}\right), \qquad \qquad \left(p>2\right). \end{equation*} More precisely, we show the uniqueness and existence of solution $\bold{u}$ and investigate a priori $L^{\infty}$ boundedness of the gradient of the solution. Assuming that the solution decays quickly at infinity, we also prove that the component $u^l$, $\left(1\leq l\leq k\right)$, converges to the function $c^l\mathcal{B}$ in space as $t\to\infty$. Here, the function $\mathcal{B}$ is the fundamental or Barenblatt solution of $p$-Laplacian equation and the constant $c^l$ is determined by the $L^1$-mass of $u^l$. The proof is based on the existence of entropy functional.\\ \indent As an application of the asymptotic large time behaviour, we establish a Harnack type inequality which makes the size of spatial average being controlled by the value of solution at one point., Comment: 23p

We establish the Krylov Safonov Harnack inequalities and Holder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal Pucci operators on manifolds that give rise to the concept of non-divergence form operators. We then provide the uniform regularity results for these operators which recover the classical results for second order local operators as limits.

This article is concerned with uniform $C^{1,\alpha}$ and $C^{1,1}$ estimates in periodic homogenization of fully nonlinear elliptic equations. The analysis is based on the compactness method, which involves linearization of the operator at each approximation step. Due to the nonlinearity of the equations, the linearized operators involve the Hessian of correctors, which appear in the previous step. The involvement of the Hessian of the correctors deteriorates the regularity of the linearized operator, and sometimes even changes its oscillating pattern. These issues are resolved with new approximation techniques, which yield a precise decomposition of the regular part and the irregular part of the homogenization process, along with a uniform control of the Hessian of the correctors in an intermediate level. The approximation techniques are even new in the context of linear equations. Our argument can be applied not only to concave operators, but also to certain class of non-concave operators., Comment: 43 pages; to be published in Arch. Rational Mech. Anal

We establish the generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear elliptic equations with rough kernels of variable orders. In contrast to the fractional Laplacian type operators having a fixed order of differentiability $\sigma \in (0,2)$, the operators under consideration have variable orders of differentiability. Since the order is not characterized by a single number, we consider a function $\varphi$ describing the variable orders of differentiability, which is allowed to oscillate between two functions $r^{\sigma_1}$ and $r^{\sigma_2}$ for some $0 < \sigma_1 \leq \sigma_2 < 2$. By introducing the generalized H\"older spaces, we provide $C^{\varphi\psi}$ estimates that generalizes the standard Evans--Krylov and Schauder type $C^{\sigma+\alpha}$ estimates.

We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $\Omega\subset \mathbb{R}^n$, $\alpha>0$ and $0

In this paper, we consider the solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ of the generalized parabolic system \begin{equation*} \left(u^i\right)_t=\nabla\cdot\left(mU^{m-1}\mathcal{A}\left(\nabla u^i,u^i,x,t\right)+\mathcal{B}\left(u^i,x,t\right)\right), \qquad \left(1\leq i\leq k\right) \end{equation*} in the range of exponents $m>\frac{n-2}{n}$ where the diffusion coefficient $U$ depends on the components of the solution $\bold{u}$. Under suitable structure conditions on the vector fields $\mathcal{A}$ and $\mathcal{B}$, we first show the uniform $L^{\infty}$ bound of the function $U$ for $t\geq \tau>0$ and law of $L^1$ mass conservation of each component $u^i$, $(i=1,\cdots,k)$, with system version of Harnack type inequality. As the last result, we also deal with the local continuity of solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ with the intrinsic scaling. If the ratio between $U$ and components $u^i$, $(i=1,\cdots,k)$, is uniformly bounded above and below, all components of the solution $\bold{u}$ have the same modulus of continuity., Comment: 40p

We investigate the evolution of population density vector, $\bold{u}=\left(u^1,\cdots,u^k\right)$, of $k$-species whose diffusion is controlled by its absolute value $\left|\bold{u}\right|$. More precisely we study the properties and asymptotic large time behaviour of solution $\bold{u}=\left(u^1,\cdots,u^k\right)$ of degenerate parabolic system \begin{equation*} \left(u^i\right)_t=\nabla\cdot\left(\left|\bold{u}\right|^{m-1}\nabla u^i\right) \qquad \mbox{for $m>1$ and $i=1,\cdots,k$}. \end{equation*} Under some regularity assumption, we prove that the function $u^i$ which describes the population density of $i$-th species with population $M_i$ converges to $\frac{M_i}{\left|\bold{M}\right|}\mathcal{B}_{\left|\bold{M}\right|}$ in space with two different approaches where $\mathcal{B}_{\left|\bold{M}\right|}$ is the Barenblatt solution of the porous medium equation with $L^1$-mass $\left|\bold{M}\right|=\sqrt{M_1^2+\cdots+M_k^2}$. \indent As an application of the asymptotic behaviour, we establish a suitable harnack type inequality which makes the spatial average of $u^i$ under control by the value of $u^i$ at one point. We also find an 1-directional travelling wave type solutions and the properties of solutions which has travelling wave behaviour at infinity., Comment: 30 pages

We study the near-the-interface behavior of a compact convex scalar curvature flow with a flat side. Under suitable initial conditions on the flat side, we show that the interface propagates with a finite and non-degenerate speed until the flat side vanishes. Then we get optimal derivative estimates of the pressure-like function, optimal decay estimates of curvatures near the interface, and an Aronson-B\'enilan-type curvature lower bound, from which we obtain the H\"older regularity of the ratio of the curvature to the optimal decay rate up to the free boundary. In the end, we obtain the short-time and all-time existence of the solution, smooth up to the interface., Comment: 35 pages

We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in \cite{CS}. Since the order of differentiability of the kernel is not characterized by a single number, we use the constant \begin{align*} C_\varphi = \left( \int_{\mathbb{R}^n} \frac{1-\cos y_1}{\vert y \vert^n \varphi (\vert y \vert)} \, dy \right)^{-1} \end{align*} instead of $2-\sigma$, where $\varphi$ satisfies a weak scaling condition. We obtain the uniform Harnack inequality and H\"older estimates of viscosity solutions to the nonlinear integro-differential equations.

In this paper, we prove the existence and uniqueness of $W^{2,p}$ ($n

We study the boundary regularity of solutions of the Dirichlet problem for the nonlocal operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized H\"older space. We prove that there exists a unique viscosity solution of $Lu = f$ in $D$, $u=0$ in $\mathbb{R}^n \setminus D$, where $D$ is a bounded $C^{1,1}$ open set, and that the solution $u$ satisfies $u \in C^V(D)$ and $u/V(d_D) \in C^\alpha (D)$ with the uniform estimates, where $V$ is the renewal function and $d_D(x) = \mbox{dist}(x, \partial D)$.

We study the local H\"older continuity and the asymptotic behaviour of solution, $\mathbf{u}=(u^1,\cdots, u^k)$, of the degenerate system \begin{equation*} u^i_t=\nabla\cdot\left(m\,U^{m-1}\nabla u^i\right) \qquad \text{for $m>1$ and $i=1,\cdots,k$ } \end{equation*} which describes the populations density of $k$-species whose diffusion is determined by their total population density $U=u^1+\cdots+u^k$. For the local H\"older continuity, we adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and Dibenedetto. Under some regularity conditions, we also prove that the population density function of $i$-th species with the population $M_i$ converges in $C_s^{\infty}$ to $\frac{M_i}{M}\mathcal{B}_M(x,t)$ as $t\to \infty$ where $\mathcal{B}_M$ is the Barenblatt profile of the standard porous medium equation with $L^1$ mass $M=M_1+\cdots+M_k$. As a consequence of asymptotic behaviour, it is shown that each density function becomes a concave function after a finite time., Comment: 37p

We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and H\"{o}lder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is H\"{o}lder continuous., Comment: 19 pages

In this paper, we establish the higher order convergence rates in periodic homogenization of viscous Hamilton-Jacobi equations, which is convex and grows quadratically in the gradient variable. We observe that although the nonlinear structure governs the first order approximation, the nonlinear effect is absorbed as an external source term of a linear equation in the second and higher order approximation. Moreover, we find that the geometric shape of the initial data has to be chosen carefully according to the effective Hamiltonian, in order to achieve the higher order convergence rates.

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, \begin{equation*} \operatorname{div}(A_s(x,u)\nabla u)=\operatorname{div}{\vec f}(x), \end{equation*} where $A_s(x,u)$ has "broken" derivatives of order $s\geq 0$, such as \begin{equation*} A_s(x,u) = a(x) + b(x)(u^+)^s, \end{equation*} with $(u^+)^0$ being understood as the characteristic function on $\{u>0\}$. The vector ${\vec f}(x)$ is assumed to be $C^\alpha$ in case $s=0$, and $C^{1,\alpha}$ (or higher) in case $s>0$. Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More exactly, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs.

We prove the natural weighted Calder\'{o}n and Zygmund estimates for solutions to elliptic and parabolic obstacle problems in nondivergence form with discontinuous coefficients and irregular obstacles. We also obtain Morrey regularity results for the Hessian of the solutions and H\"{o}lder continuity of the gradient of the solutions., Comment: 18 pages

In this paper, we prove local $C^{1}$ regularity of free boundaries for the double obstacle problem with an upper obstacle $\psi$, \begin{align*} \Delta u &=f\chi_{\Omega(u) \cap\{ u< \psi\} }+ \Delta \psi \chi_{\Omega(u)\cap \{u=\psi\}}, \qquad u\le \psi \quad \text { in } B_1, \end{align*} where $\Omega(u)=B_1 \setminus \left( \{u=0\} \cap \{ \nabla u =0\}\right)$ under a thickness assumption for $u$ and $\psi$., Comment: 21 pages

We establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic Cauchy problems accompanied with rapidly oscillating initial data. Such result is new even for linear problems. Here we construct higher order initial layer and interior correctors, which describe the oscillatory behavior near the initial and interior time zone of the domain. To construct higher order correctors, we develop a regularity theory in macroscopic scales, and prove an exponential decay estimate for initial layer correctors. The higher order expansion requires an iteration process: successively correcting the initial layer, then the interior. This leads to a more complicated asymptotic expansion, as compared to the non-oscillating data case, and this complexity is present even in the linear case. A notable observation for fully nonlinear operators is that even if the given operator is space-time periodic, the interior correctors become aperiodic in the time variable as we proceed with the iteration process. Moreover, each interior corrector of higher order is paired with a space-time periodic version, and the difference between the two decays exponentially fast with time.

We derive local $C^{2}$ estimates for complete non-compact translating solitons of the Gauss curvature flow in $\mathbb{R}^3$ which are graphs over a convex domain $\Omega$. This is closely is related to deriving local $C^{1,1}$ estimates for the degenerate Monge-Amp\'ere equation. As a result, given a weakly convex bounded domain $\Omega$, we establish the existence of a $C^{1,1}_{\text{loc}}$ translating soliton. In particular, when the boundary $\partial \Omega$ has a line segment, we show the existence of flat sides of the translator from a local a'priori non-degeneracy estimate near the free-boundary.