arXiv subjects
Publications and source records attributed to Yuzhou Fang.
We consider a class of generalized nonlocal $p$-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates.
We consider the nonlocal double phase equation \begin{align*} \mathrm{P.V.} &\int_{\mathbb{R}^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{sp}(x,y)\,dy\\ &+\mathrm{P.V.} \int_{\mathbb{R}^n} a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{tq}(x,y)\,dy=0, \end{align*} where $1<p\leq q$ and the modulating coefficient $a(\cdot,\cdot)\geq0$. Under some suitable hypotheses, we first use the De Giorgi-Nash-Moser methods to derive the local Hölder continuity for bounded weak solutions, and then establish the relationship between weak solutions and viscosity solutions to such equations.
We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $$ \partial_t u=[|D u|^q+a(x,t)|D u|^s]\left(Δu+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D u|}\right\rangle\right), $$ where $1<p<\infty$, $-1<q\leq s<\infty$ and $a(x,t)\ge 0$. The motivation to investigate this model stems not only from the connections to tug-of-war like stochastic games with noise, but also from the non-standard growth problems of double phase type. According to different values of $q,s$, such equations include nonhomogeneous degeneracy or singularity, and may involve these two features simultaneously. In particular, when $q=p-2$ and $q<s$, it will encompass the parabolic $p$-Laplacian both in divergence form and in non-divergence form. We aim to explore the from $L^\infty$ to $C^{1,α}$ regularity theory for the aforementioned problem. To be precise, under some proper assumptions, we use geometrical methods to establish the local Hölder regularity of spatial gradients of viscosity solutions.
We consider the fully nonlinear equation with variable-exponent double phase type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$ Under some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques we obtain the sharp local $C^{1,α}$ regularity of viscosity solutions to such equations.
We investigate the different notions of solutions to the double-phase equation $$ -\dive(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=0, $$ which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the $\mathcal{A}_{H(\cdot)}$-harmonic functions of nonlinear potential theory, and then show that $\mathcal{A}_{H(\cdot)}$-harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.
We consider interior Hölder regularity of the spatial gradient of viscosity solutions to the normalized $p(x,t)$-Laplace equation $$ u_t=\left(δ_{ij}+(p(x,t)-2)\frac{u_i u_j}{|Du|^2}\right)u_{ij} $$ with some suitable assumptions on $p(x,t)$, which arises naturally from a two-player zero-sum stochastic differential game with probabilities depending on space and time.