Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotic Mean Value Expansions for Solutions of General Second-Order Elliptic Equations
87
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We obtain asymptotic mean value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. We study both when the set of coefficients is bounded and unbounded (each case requires different techniques). The families of equations that we consider include well-known operators such as Pucci, Issacs, and $k-$Hessian operators.
rate research
Read More
We consider a boundary value problem in a bounded domain involving a degenerate operator of the form $$L(u)=-textrm{div} (a(x) abla u)$$ and a suitable nonlinearity $f$. The function $a$ vanishes on smooth 1-codimensional submanifolds of $Omega$ where it is not allowed to be $C^{2}$. By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where $a$ vanishes.
In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ partial_t u+(-Delta)^{frac{theta}{2}}u=0quadmbox{in}quad{bf R}^Ntimes(0,infty), qquad u(x,0)=varphi(x)quadmbox{in}quad{bf R}^N, $$ where $0<theta<2$ and $varphiin L_K:=L^1({bf R}^N,,(1+|x|)^K,dx)$ with $Kge 0$. Furthermore, we develop the arguments in [15] and [18] and establish a method to obtain the asymptotic expansions of the solutions to a nonlinear fractional diffusion equation $$ partial_t u+(-Delta)^{frac{theta}{2}}u=|u|^{p-1}uquadmbox{in}quad{bf R}^Ntimes(0,infty), $$ where $0<theta<2$ and $p>1+theta/N$.
This paper concerns about the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumption which called {sl basic assumptions}, but also some technical assumptions which we called {sl further assumptions}. It is shown as usual by first applying the Holmgren transform to this inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate given via a partition of unity and Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.
In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge-Amp`ere equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of Dynamic Programming Principles for certain two-player, zero-sum games.
We use blow up analysis for local integral equations to prove compactness of solutions to higher order critical elliptic equations provided the potentials only have non-degenerate zeros. Secondly, corresponding to Schoens Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow up points of solutions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
