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Register a new userDirichlet forms and degenerate elliptic operators
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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 $A$.
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In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated to second order linear degenerate elliptic partial differential operators $X$, with rough coefficients of the form $$X=-text{div}(P abla )+{bf HR}+{bf S^prime G} +F$$ in a geometric homogeneous space setting where the $ntimes n$ matrix function $P=P(x)$ is allowed to degenerate. We give a maximum principle for weak solutions of $Xuleq 0$ and follow this with a result describing a relationship between compact projection of the degenerate Sobolev space $QH^{1,p}$ into $L^q$ and a Poincare inequality with gain adapted to $Q$.
We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $mathbb{R}^d_+$, where the coefficients are the product of $x_d^alpha, alpha in (-infty, 1),$ and a bounded uniformly elliptic matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary ${x_d =0}$ and they may not locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations.
In this paper we study the existence of solutions of thedegererate elliptic system.
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 $L$.
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.
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