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Register a new userA note on estimates of level sets and their role in demonstrating regularity of solutions to nonlocal double phase equations
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In this note we prove an estimate on the level sets of a function with $(p, q)$ growth that depends on the difference quotient of a bounded weak solution to a nonlocal double phase equation. This estimate is related to a self improving property of these solutions.
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We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic ``phases that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work ``Nonlocal self-improving properties Anal. PDE, 8(1):57--114 for the specific nonlinear setting under investigation in this manuscript.
This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - mbox{div}[mathbb{A}(x,t,u, abla u)]= mbox{div}[{mathbf F}]$ with homogeneous Dirichlet boundary conditions over bounded spatial domains. Our main focus is on the case that the vector coefficients $mathbb{A}$ are discontinuous and singular in $(x,t)$-variables, and dependent on the solution $u$. Global and interior weighted $W^{1,p}(Omega, omega)$-regularity estimates are established for weak solutions of these equations, where $omega$ is a weight function in some Muckenhoupt class of weights. The results obtained are even new for linear equations, and for $omega =1$, because of the singularity of the coefficients in $(x,t)$-variables
We show that local weak solutions to parabolic systems of p-Laplace type are H{o}lder continuous in time with values in a spatial Lebesgue space and H{o}lder continuous on almost every time line. We provide an elementary and self-contained proof building on the local higher integrability result of Kinnunen and Lewis.
This paper is concerned with a class of nonlocal dispersive models -- the $theta$-equation proposed by H. Liu [ On discreteness of the Hopf equation, {it Acta Math. Appl. Sin.} Engl. Ser. {bf 24}(3)(2008)423--440]: $$ (1-partial_x^2)u_t+(1-thetapartial_x^2)(frac{u^2}{2})_x =(1-4theta)(frac{u_x^2}{2})_x, $$ including integrable equations such as the Camassa-Holm equation, $theta=1/3$, and the Degasperis-Procesi equation, $theta=1/4$, as special models. We investigate both global regularity of solutions and wave breaking phenomena for $theta in mathbb{R}$. It is shown that as $theta$ increases regularity of solutions improves: (i) $0 <theta < 1/4$, the solution will blow up when the momentum of initial data satisfies certain sign conditions; (ii) $1/4 leq theta < 1/2$, the solution will blow up when the slope of initial data is negative at one point; (iii) ${1/2} leq theta leq 1$ and $theta=frac{2n}{2n-1}, nin mathbb{N}$, global existence of strong solutions is ensured. Moreover, if the momentum of initial data has a definite sign, then for any $thetain mathbb{R}$ global smoothness of the corresponding solution is proved. Proofs are either based on the use of some global invariants or based on exploration of favorable sign conditions of quantities involving solution derivatives. Existence and uniqueness results of global weak solutions for any $theta in mathbb{R}$ are also presented. For some restricted range of parameters results here are equivalent to those known for the $b-$equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the b-equation, {it J. reine angew. Math.}, {bf 624} (2008)51--80.]
Given $Lgeq 1$, we discuss the problem of determining the highest $alpha=alpha(L)$ such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by $L$ is in $C^alpha_{rm loc}$. This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $alpha(L)gtrsim {rm exp}(-CL^beta)$, for some $C, betageq 1$ depending on the dimension $Ngeq 3$. We show that in the non-local case, $alpha(L)gtrsim L^{-1-delta}$ for all $delta>0$.
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