Brice Camus
We study the semi-classical trace formula at a critical energy level for an $h$-pseudo-differential operator on $\mathbb{R}^{n}$ whose principal symbol has a totally degenerate critical point for that energy. This problem is studied for a large time behavior and under the hypothesis that the principal symbol of the operator has a local extremum at the critical point.
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The Dirichlet problem for a family of totally degenerate differential operators
在势论框架下，证明了Bony意义下一类完全退化微分算子族相关狄利克雷问题的Perron-Weiner-Brelot-Bauer解的存在性，还给出边界点正则性的Wiener型准则和外锥条件，结果适用于含示例的一大类强退化算子。
In this paper, we analyze main properties of the volume and layer potentials as well as the Poisson integral for a multi-dimensional degenerate parabolic equation. As consequences, we obtain trace formulae of the heat volume potential and the Poisson integral which solve Kac's problem for degenerate parabolic equations in cylindrical domains.
We prove existence of strong traces at $t=0$ for quasi-solutions to (multidimensional) degenerate parabolic equations with no non-degeneracy conditions. In order to solve the problem, we combine the blow up method and a strong precompactness result for quasi-solutions to degenerate parabolic equations with the induction argument with respect to the space dimension.
We prove nonexistence results of Liouville type for nonnegative viscosity solutions of some equations involving the fully nonlinear degenerate elliptic operators ${\cal P}^\pm_k$, defined respectively as the sum of the largest and the smallest $k$ eigenvalues of the Hessian matrix. For the operator ${\cal P}^+_k$ we obtain results analogous to those which hold for the Laplace operator in space dimension $k$. Whereas, owing to the stronger degeneracy of the operator ${\cal P}^-_k$, we get totally different results.
We study the semi-classical trace formula at a critical energy level for a $h$-pseudo-differential operator whose principal symbol has a unique non-degenerate critical point for that energy. This leads to the study of Hamiltonian systems near equilibrium and near the non-zero periods of the linearized flow. The contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals. The new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface.
We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on $\mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result, which establishes the contribution of the associated equilibrium in the trace formula, is valid for all time in a compact subset of $\mathbb{R}$ and includes the singularity in $t=0$. For these new contributions the asymptotic expansion involves the logarithm of the parameter $h$. Depending on an explicit arithmetic condition on the dimension and the order of…
Hidden Boundary Trace Regularity and an Observability Estimate with Interior Remainder for Boundary-Degenerate Hyperbolic Equations
本文研究了二维退化双曲方程的隐边界迹正则性，建立了加权Sobolev空间中的well-posedness，并证明了法向导数的L^2迹估计。通过适应各向异性退化的Carleman权重，推导出具有低阶内部余项的长时间观测估计，同时识别了临界阈值处的框架级障碍。
Balian-Bloch Wave Invariants for Nearly Degenerate Orbits
本文研究平面台球的波迹奇异支集与长度谱差异，通过近简并 Poincaré 映射周期轨道推导 Balian–Bloch 正则化预解式渐近迹公式，并分析 Maslov 指标与系数抵消。
We study the initial value problem with unbounded nonnegative functions or measures for the equation $ \prt_tu-\Gd_p u+f(u)=0$ in $\BBR^N\ti(0,\infty)$ where $p>1$, $\Gd_p u = \text{div}(\abs {\nabla u}^{p-2} \nabla u)$ and $f$ is a continuous, nondecreasing nonnegative function such that $f(0)=0$. In the case $p>\frac{2N}{N+1}$, we provide a sufficient condition on $f$ for existence and uniqueness of the solutions satisfying the initial data $k\gd_0$ and we study their limit when $k\to\infty$ according $f^{-1}$ and $F^{-1/p}$ are integrable or not at infinity, where $F(s)=\int_0^s f(\gs)d\gs$…
We consider a mixed type boundary value problem for a class of degenerate parabolic-hyperbolic equations. Namely, we consider a Cartesian product domain and split its boundary into two parts. In one of them we impose a Dirichlet boundary condition; in the other, we impose a Neumann condition. We apply a normal trace formula for $L^2$-divergence-measure fields to prove a new strong trace property in the part of the boundary where the Neumann condition is imposed. We prove existence and uniqueness of the entropy solution. This is a revised corrected version of the paper published in Arch.\ Ratio…
We establish spectral estimates at a critical energy level for $h$-pseudors . Via a trace formula, we compute the contribution of isolated (non-extremum) critical points under a condition of "real principal type". The main result holds for all dimensions, for a singularity of any finite order and can be invariantly expressed in term of the geometry of the singularity. When the singularities are not integrable on the energy surface the results are significative since the order w.r.t. $h$ of the spectral distributions are bigger than in the regular setting.
We give sufficient conditions for the existence and uniqueness, in bounded uniformly convex domains $Ω$, of solutions of degenerate elliptic equations depending also on the nonlinear gradient term $H$, in term of the size of $Ω$, of the forcing term $f$ and of $H$. The results apply to a wide class of equations, having as principal part significant examples, e.g. linear degenerate operators, weighted partial trace operators and the homogeneous Monge-Ampère operator.
Sobolev--Morrey Spaces and Divergence-Form Degenerate Second-Order Elliptic Equations on Domains with Higher Co-Dimensional Boundaries
研究\(\mathbb{R}^n\)中高余维边界区域上加权齐次Sobolev - Morrey空间，建立实变理论，应用迹和延拓算子有界性，获散度型退化二阶椭圆方程Dirichlet问题加权先验估计。
Non-homogeneous boundary value problems for second-order degenerate hyperbolic equations and their application
本文研究了退化双曲方程的非齐次边界值问题，提出了适用于退化情形的解理论，并推广了希尔伯特唯一性方法到退化波方程。
Shape Design for a Class of Degenerate Parabolic Equations with Boundary Point Degeneracy and Its Application to Boundary Observability
本文研究退化抛物方程的边界点退化问题，通过形状设计方法分析边界观测问题，提出高维抛物方程的形状设计方法，建立退化方程的观测不等式。
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