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Register a new userGreen kernel and Martin kernel of Schrodinger operators with singular potential and application to the B.V.P. for linear elliptic equations
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Let $Omega subset mathbb{R}^N$ ($N geq 3$) be a $C^2$ bounded domain and $K subset Omega$ be a compact, $C^2$ submanifold in $mathbb{R}^N$ without boundary, of dimension $k$ with $0leq k < N-2$. We consider the Schrodinger operator $L_mu = Delta + mu d_K^{-2}$ in $Omega setminus K$, where $d_K(x) = text{dist}(x,K)$. The optimal Hardy constant $H=(N-k-2)/2$ is deeply involved in the study of $-L_mu$. When $mu leq H^2$, we establish sharp, two-sided estimates for Green kernel and Martin kernel of $-L_mu$. We use these estimates to prove the existence, uniqueness and a priori estimates of the solution to the boundary value problem with measures for linear equations associated to $-L_mu$
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Let $Omega subset {mathbb R}^N$ ($N geq 3$) be a $C^2$ bounded domain and $F subset partial Omega$ be a $C^2$ submanifold of dimension $0 leq k leq N-2$. Put $delta_F(x)=dist(x,F)$, $V=delta_F^{-2}$ in $Omega$ and $L_{gamma V}=Delta + gamma V$. Denote by $C_H(V)$ the Hardy constant relative to $V$ in $Omega$. We study positive solutions of equations (LE) $-L_{gamma V} u = 0$ and (NE) $-L_{gamma V} u+ f(u) = 0$ in $Omega$ when $gamma < C_H(V)$ and $f in C({mathbb R})$ is an odd, monotone increasing function. We establish the existence of a normalized boundary trace for positive solutions of (LE) - first studied by Marcus and Nguyen for the case $F=partial Omega$ - and employ it to investigate the behavior of subsolutions and super solutions of (LE) at the boundary. Using these results we study boundary value problems for (NE) and derive a-priori estimates. Finally we discuss subcriticality of (NE) at boundary points of $Omega$ and establish existence and stability results when the data is concentrated on the set of subcritical points.
Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients.
We study the Cauchy problem for the generalized elliptic and non-elliptic derivative nonlinear Schrodinger equations, the existence of the scattering operators and the global well posedness of solutions with small data in Besov spaces and in modulation spaces are obtained. In one spatial dimension, we get the sharp well posedness result with small data in critical homogeneous Besov spaces. As a by-product, the existence of the scattering operators with small data is also shown. In order to show these results, the glob
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 $alpha =0$
We introduce a vector differential operator $mathbf{P}$ and a vector boundary operator $mathbf{B}$ to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator $L:=mathbf{P}^{ast T}mathbf{P}$ with homogeneous or nonhomogeneous boundary conditions given by $mathbf{B}$, where we ensure that the distributional adjoint operator $mathbf{P}^{ast}$ of $mathbf{P}$ is well-defined in the distributional sense. We represent the inner product of the reproducing-kernel Hilbert space in terms of the operators $mathbf{P}$ and $mathbf{B}$. In addition, we find relationships for the eigenfunctions and eigenvalues of the reproducing kernel and the operators with homogeneous or nonhomogeneous boundary conditions. These eigenfunctions and eigenvalues are used to compute a series expansion of the reproducing kernel and an orthonormal basis of the reproducing-kernel Hilbert space. Our theoretical results provide perhaps a more intuitive way of understanding what kind of functions are well approximated by the reproducing kernel-based interpolant to a given multivariate data sample.
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