Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn Elliptic System with Degenerate Coercivity
91
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we study the existence of solutions of thedegererate elliptic system.
rate research
Read More
We study a degenerate elliptic equation, proving existence results of distributional solutions in some borderline cases.
Let $Omega subset mathbb{R}^{n+1}$ be an open set whose boundary may be composed of pieces of different dimensions. Assume that $Omega$ satisfies the quantitative openness and connectedness, and there exist doubling measures $m$ on $Omega$ and $mu$ on $partial Omega$ with appropriate size conditions. Let $Lu=-mathrm{div}(A abla u)$ be a real (not necessarily symmetric) degenerate elliptic operator in $Omega$. Write $omega_L$ for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) $omega_L in A_{infty}(mu)$, (ii) $L$ is $L^p(mu)$-solvable for some $p in (1, infty)$, (iii) every bounded null solution of $L$ satisfies a Carleson measure estimate with respect to $mu$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(mu)$ for some (or for all) $q in (0, infty)$ for any null solution of $L$, (v) $L$ is $mathrm{BMO}(mu)$-solvable, and (vi) every bounded null solution of $L$ is $varepsilon$-approximable for any $varepsilon>0$. On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of $omega_L$ with respect to $mu$ in terms of local $L^2(mu)$ estimates of the truncated conical square function for any bounded null solution of $L$. This is also equivalent to the finiteness $mu$-almost everywhere of the truncated conical square function for any bounded null solution of $L$.
It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B) mapsto d(A ;B)in[0,infty]$ over pairs of measurable subsets of $Ri^d$. Then [ |(phi_A,S_tphi_B)|leq e^{-d(A;B)^2(4t)^{-1}}|phi_A|_2|phi_B|_2 ] for all $t>0$ and all $phi_Ain L_2(A)$, $phi_Bin L_2(B)$. Moreover $S_tL_2(A)subseteq L_2(A)$ for all $t>0$ if and only if $d(A ;A^c)=infty$ where $A^c$ denotes the complement of $A$.
We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a unique W^{1,1}_0 distributional solution under suitable summability assumptions on the source in Lebesgue spaces. Moreover, we prove that our problem has no solution if the source is a Radon measure concentrated on a set of zero harmonic capacity.
We study both divergence and non-divergence form parabolic and elliptic equations in the half space ${x_d>0}$ whose coefficients are the product of $x_d^alpha$ and uniformly nondegenerate bounded measurable matrix-valued functions, where $alpha in (-1, infty)$. As such, the coefficients are singular or degenerate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the $x_d$ direction and have small mean oscillation in the other directions in small cylinders. Our results are new even in the special case when the coefficients are constants, and they are reduced to the classical results when $alpha =0$
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
