Good elliptic operators on Cantor sets

雪花的良好椭圆算子

Good elliptic operators on snowflakes

本文在Koch型雪花补集上构造了标量系数的椭圆算子，使其椭圆测度等于边界Hausdorff测度，展示了纯不可求长边界上椭圆测度行为的极端差异。

椭圆方程在弦域中Neumann问题的可解性

Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains

本文研究了在弦域中椭圆算子的Neumann问题的可解性，证明了在满足特定条件时，该问题在L^p空间中可解。

Reduced measures for semilinear elliptic equations involving Dirichlet operators

We consider elliptic equations of the form (E) $-Au=f(x,u)+μ$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $μ$ is a Borel measure of finite potential. We introduce a probabilistic definition of a solution of (E), develop the theory of good and reduced measures introduced by H. Brezis, M. Marcus and A.C. Ponce in the case where $A=Δ$ and show basic properties of solutions of (E). We also prove Kato's type inequality. Finally, we characterize the set of good measures in case $f(u)=-u^p$ for some…

Good-$λ$ type bounds of quasilinear elliptic equations for the singular case

In this paper, we study the good-$λ$ type bounds for renormalized solutions to nonlinear elliptic problem: \begin{align*} \begin{cases} -÷(A(x,\nabla u)) &= μ\quad \text{in} \ \ Ω, \\ u &=0 \quad \text{on} \ \ \partial Ω. \end{cases} \end{align*} where $Ω\subset \mathbb{R}^n$, $μ$ is a finite Radon measure and $A$ is a monotone Carathédory vector valued function defined on $W^{1,p}_0(Ω)$. The operator $A$ satisfies growth and monotonicity conditions, and the $p$-capacity uniform thickness condition is imposed on $\mathbb{R}^n \setminus Ω$, for the singular case $\frac{3n-2}{2n-1} < p \le 2- \f…

Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces

We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation $-{\rm div}(A\nabla u+bu)+c\nabla u+du=-{\rm div}f+g$, assuming that $A$ is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions $b,f\in L^{n,1}$, $d,g\in L^{\frac{n}{2},1}$ and $c\in L^{n,q}$ for $q\leq\infty$, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with "good" constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary sma…

Buffon's needle landing near Besicovitch irregular self-similar sets

In this paper we get an estimate of Favard length of an arbitrary neighbourhood of an arbitrary self-similar Cantor set. Consider $L$ closed disjoint discs of radius $1/L$ inside the unit disc. By using linear maps of smaller disc onto the unit disc we can generate a self-similar Cantor set $G$. Then $\G=\bigcap_n\G_n$. One may then ask the rate at which the Favard length - the average over all directions of the length of the orthogonal projection onto a line in that direction - of these sets $\G_n$ decays to zero as a function of $n$. The quantitative results for the Favard length problem wer…