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Register a new userOptimal elliptic regularity: a comparison between local and nonlocal equations
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Authors
Sunra Mosconi
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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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This paper proves Holder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first shows that viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative are Holder continuous.
In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $displaystyle -operatorname{div}(A(| abla u|) abla u)+Bleft( | abla u|right) =f(u)$; in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
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.]
Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) ${mathbb L} u = u^p +lambda mu$ in a bounded domain $Omega$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $lambda>0$. The operator ${mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $mu$ is taken in the optimal weighted measure space. The interplay between the operator ${mathbb L}$, the source term $u^p$ and the datum $mu$ yields substantial difficulties and reveals the distinctive feature of the problem. We develop a new unifying technique based on a fine analysis on the Green kernel, which enables us to construct a theory for semilinear equation (E) in measure frameworks. A main thrust of the paper is to provide a fairly complete description of positive solutions to the Dirichlet problem for (E). In particular, we show that there exist a critical exponent $p^*$ and a threshold value $lambda^*$ such that the multiplicity holds for $1<p<p^*$ and $0<lambda<lambda^*$, the uniqueness holds for $1<p<p^*$ and $lambda=lambda^*$, and the nonexistence holds in other cases. Various types of nonlocal operator are discussed to exemplify the wide applicability of our theory.
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