Strongly elliptic operators with distributional coefficients

Schroedinger and elliptic operators with distributional coefficients on a bounded domain

We study the operator $L=-Δ+q$ on a bounded domain $Ω\subset\mathbb R^n$, where $q(x)$ is a distributional potential. We find sufficient conditions for $q(x)$ which guarantee that $L$ is well--defined with Dirichlet and generalized Neumann boundary conditions. The asymptotics of the eigenvalues and the basis properties of the eigen- and associated functions of such operators are studied. The results are generalized for strongly elliptic operator of order $2m$ with Dirichlet boundary conditions.

Quasilinear elliptic equations and weighted Sobolev-Poincaré inequalities with distributional weights

We introduce a class of weak solutions to the quasilinear equation $-Δ_p u = σ|u|^{p-2}u$ in an open set $Ω\subset\mathbf{R}^n$. Here $p>1$, and $Δ_p u$ is the $p$-Laplacian operator. Our notion of solution is tailored to general distributional coefficients $σ$ satisfying a certain weighted Sobolev-Poincare inequality. We also study weak solutions of the closely related equation $-Δ_p v = (p-1)|\nabla v|^p + σ$, under the same conditions on $σ$. Our results for this latter equation will allow us to characterize the class of distributions $σ$ which satisfy the Sobolev-Poincare inequality, there…

Spectral flow and bifurcation for a class of strongly indefinite elliptic systems

We consider bifurcation of solutions from a given trivial branch for a class of strongly indefinite elliptic systems via the spectral flow. Our main results establish bifurcation invariants that can be obtained from the coefficients of the systems without using explicit solutions of their linearisations at the given branch. Our constructions are based on a comparison principle for the spectral flow and a generalisation of a bifurcation theorem due to Szulkin.

A nonsmooth variational approach to differential problems. A case study of nonresonance under the first eigenvalue for a strongly nonlinear elliptic problem

We adapt a technique of nonsmooth critical point theory developed by Degiovanni-Zani for a semilinear problem involving the Laplacian to the the case of the $p$-Laplacian. We suppose only coercivity conditions on the potential and impose no growth condition of the nonlinearity. The coercivity is obtained using similar nonresonance conditions to [Mawhin-Ward-Willem] and to [Landesman-Lazer] in two different results and using some comparison functions and comparison spaces in a third one. It is also shown that neither of the three theorems implies the two others.

分布分数空间中的莱布尼茨法则与高斯-格林公式

Leibniz rules and Gauss-Green formulas in distributional fractional spaces

本文基于分布方法，在分数阶函数空间中证明了新的莱布尼茨法则，并将其应用于一般分数阶椭圆算子边值问题的适定性证明。

具有分布系数的椭圆算子的定义域

On domains of elliptic operators with distributional coefficients

利用奇异性SPDE和Paracontrolled分布理论，构造奇异椭圆算子的定义域，并证明正则化算子的范数预解收敛性，应用于奇异薛定谔方程的局部适定性。