Degenerate Elliptic Equations

We study the regularity of weak solutions for a class of second order semi-linear infinitely degenerate elliptic equations. We get the regularity of weak solutions up to the boundary for Dirichlet problem, by noting the logarithmic regularity estimate for a linear principal part. In relation to this linear part, we also show the controllability and strong maximum principle for second order hypoelliptic operators even in the case where they degenerate infinitely. Model equations naturally come from some variational problems, if one replace the Laplace operator in such as Yamabe problems by degenerate elliptic operators. In the infinitely degenerate case, a permissible nonlinear term is not fractional power, compared with elliptic or subelliptic case. To treat this nonlinear term, the nonlinear microlocal analysis is developed in the logarithmic Sobolev space.

Spectral asymptotics of a class of non-self-adjoint degenerate elliptic differential operators

Continuity properties of solutions to some degenerate elliptic equations

σ/|α|=|β|=m Dα(a αβDβ) is a higher order degenerate-elliptic operator on L2(RN) and V ∈ L1(RN) we show in particular that s(λ) > 0 for all λ > 0 provided that ∫RN V(x)dx≥ 0, V not equivalent to 0 and N ≤ 2m.

The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable is , where A is a finitely many orders smoothing linear hypoelliptic operator and is the noise magnitude. The covariance operator CU of U is smoothing of order , self-adjoint, injective and elliptic pseudodifferential operator. If was taking values in L2 then in Gaussian case solving the conditional mean (and maximum a posteriori) estimate is linked to solving the minimisation problem However, Gaussian white noise does not take values in L2 but in where is big enough. A modification of the above approach to solve the inverse problem is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of the conditional mean estimate to the correct solution as is proven in appropriate function spaces using microlocal analysis. Also the frequentist posterior contractions rates are studied.

We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms under very general hypotheses. Our results apply to a wide class of strongly elliptic, subelliptic and degenerate elliptic operators. In the degenerate case the asymptotics incorporate possible non-ergodicity.

The spaces ${\mathcal {K}_{mk}}$, introduced in the Nehari Volume of Journal d’Analyse Mathématique, for nonnegative integer values of $m$ and arbitrary real values of $k$ are extended to negative values of $m$. The extension is consistent with the equivalence $\parallel {\zeta ^j}u{\parallel _{m,k}}\sim \parallel u{\parallel _{m,k - j}}$, the inequality $\parallel {D^\alpha }u{\parallel _{m,k}} \leqslant {\text {const}}\parallel u{\parallel _{m + |\alpha |,k + |\alpha |}}$, and the generalized Cauchy-Schwarz inequality $|\langle {u,v} \rangle | \leqslant \parallel u {\parallel _{m,k}}\parallel v\parallel _{ - m, - k}$. (Here $\langle u, \upsilon \rangle$ is the ${L_2}$ scalar product.) There exists a second order degenerate elliptic operator which maps ${\mathcal {K}_{m,k}} 1 - 1$ onto ${\mathcal {K}_{m - 2,k}}$. These facts are used to simplify proof of regularity theorems for elliptic and hyperbolic problems and to give new results concerning growth rates at the boundary for the coefficients of the operator and the forcing function. (See Notices Amer. Math. Soc.

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Local Hopf Lemma for Certain Degenerate Elliptic Operators on the Half-Space

Fully nonlinear degenerate equations with sublinear gradient term

On the regularity of solutions to a class of degenerate PDE's with lower order terms

General Classes of Degenerate Elliptic Differential Operators

The calculus of classical pseudo-differential operators has been used in a fundamental way in the study of boudary value problems associated to (systems) of elliptic differential and pseudo-differential operators. This calculus was used by Hörmander in [7] to construct parametrices for the elliptic operators, using which the boundary value problem is reduced to a system of pseudo-differential operators on the boundary. In order to treat the boundary value problems for parabolic operators in bounded cylindrical domains Piriou introduced in [15] and [16] a class of operators which called pseudo-differential operators of Volterra type. The basic idea consists in developping a calculus of an appropriate class of anisotropic pseudo-differential operators as in an earlier note of Hunt and Piriou [8].KeywordsVector BundleParabolic SystemPrincipal SymbolCylindrical DomainVolterra TypeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this manuscript we investigate the regularization of solutions for the spatially homogeneous Landau equation. For moderately soft potentials, it is shown that weak solutions become smooth instantaneously and stay so over all times, and the estimates depend only on the initial mass, energy, and entropy. For very soft potentials we obtain a conditional regularity result, hinging on what may be described as a nonlinear Morrey space bound, assumed to hold uniformly over time. This bound always holds in the case of very soft potentials, and nearly holds for general potentials, including Coulomb. This latter phenomenon captures the intuition that for very soft potentials, the dissipative term in the equation is of the same order as the quadratic term driving the growth (and potentially, singularities). In particular, for the Coulomb case, the conditional regularity result shows a rate of regularization much stronger than what is usually expected for regular parabolic equations. The main feature of our proofs is the analysis of the linearized Landau operator around an arbitrary and possibly irregular distribution. This linear operator is shown to be a degenerate elliptic Schrodinger operator whose coefficients are controlled by $$A_p$$ -weights.

The Quantization of Edge Symbols.- On Rays of Minimal Growth for Elliptic Cone Operators.- Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds.- Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups.- On the Fourier Analysis of Operators on the Torus.- Wave Kernels of the Twisted Laplacian.- Super-exponential Decay of Solutions to Differential Equations in ?d.- Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order.- A New Aspect of the L p-extension Problem for Inhomogeneous Differential Equations.- Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols.- Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces.- Algebras of Pseudo-differential Operators with Discontinuous Symbols.- A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform.- A Characterization of Stockwell Spectra.- Exact and Numerical Inversion of Pseudo-differential Operators and Applications to Signal Processing.- On the Product of Localization Operators.- Gelfand-Shilov Spaces, Pseudo-differential Operators and Localization Operators.- Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces.- Microlocalization within Some Classes of Fourier Hyperfunctions.