Degenerate Elliptic Equations Research Articles

Some auxiliary estimates for solutions to non-uniformly degenerate second-order elliptic equations

We consider a class of second order elliptic equations in divergence form with non-uniform exponential degeneracy. The method used is based on the fact that the degeneracy rates of the eigenvalues of the matrix ||aij(x)|| (function λi(x)) are not the functions of unusual norm |x|, but of some anisotropic distance |x| a−. We assume that the Dirichlet problem for such equations is solvable in the classical sense for every continuous boundary function in any normal domain Ω. Estimates for the weak solutions of Dirichlet problem near the boundary point are obtained, and Green’s functions for second order non-uniformly degenerate elliptic equations are constructed.

The impact of a singular first-order term in some degenerate elliptic equations involving Hardy potential

The impact of a singular first-order term in some degenerate elliptic equations involving Hardy potential

Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization

AbstractWe consider degenerate elliptic equations of second order in divergence form with a symmetric random coefficient field a. Extending the work of Bella et al. (Ann Appl Probab 28(3):1379–1422, 2018), who established the large-scale $$C^{1,\alpha }$$ C 1 , α regularity of a-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius $$r_*$$ r ∗ describing the minimal scale for this $$C^{1,\alpha }$$ C 1 , α regularity. As an application to stochastic homogenization, we partially generalize results by Gloria et al. (Anal PDE 14(8):2497–2537, 2021) on the growth of the corrector, the decay of its gradient, and a quantitative two-scale expansion to the degenerate setting. On a technical level, we demand the ensemble of coefficient fields to be stationary and subject to a spectral gap inequality, and we impose moment bounds on a and $$a^{-1}$$ a - 1 . We also introduce the ellipticity radius $$r_e$$ r e which encodes the minimal scale where these moments are close to their positive expectation value.

On the Weak Solvability of Dirichlet Problem for a Fractional Order Degenerate Elliptic Equation

On the Weak Solvability of Dirichlet Problem for a Fractional Order Degenerate Elliptic Equation

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

<p style='text-indent:20px;'>We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups.</p>

On higher order boundary Harnack and analyticity of free boundaries

We establish a C1,α Schauder estimate of a non-standard degenerate elliptic equation and use it to give another proof of the higher order boundary Harnack inequality. As an application, we obtain the analyticity of the free boundary in the classical obstacle problem based on iterating the boundary Harnack principle.

Solving stationary inverse heat conduction in a thin plate

We consider a steady state heat conduction problem in a thin plate. In the application, it is used to connect two cylindrical containers and fix their relative positions. At the same time it serves to measure the temperature on the inner cylinder. We derive a two dimensional mathematical model, and use it to approximate the heat conduction in the thin plate. Since the plate has sharp edges on the sides the resulting problem is described by a degenerate elliptic equation. To find the temperature in the interior part from the exterior measurements, we formulate the problem as a Cauchy problem for stationary heat equation. We also reformulate the Cauchy problem as an operator equation, with a compact operator, and apply the Landweber iteration method to solve the equation. The case of the degenerate elliptic equation has not been previously studied in this context. For numerical computation, we consider the case where noisy data is present and analyse the convergence.

A Perspective on the Regularity Theory of Degenerate Elliptic Equations

A Perspective on the Regularity Theory of Degenerate Elliptic Equations

Semilinear degenerate elliptic equation in the presence of singular nonlinearity

Semilinear degenerate elliptic equation in the presence of singular nonlinearity

Existence of the solution for a class of the semilinear degenerate elliptic equation involving the Grushin operator in R2$\mathbb {R}^2$: The interaction between Grushin's critical exponent and exponential growth

AbstractIn this work, we study the existence of the solution for a class of semilinear degenerate elliptic equations in the whole space involving the Grushin operator and a nonlinearity that involves the Grushin's critical growth and the exponential growth. The existence of at least one nontrivial weak solution is done by combining the mountain‐pass theorem, Trudinger–Moser inequality, and a version of a result due to Lions for the exponential growth in in the corresponding weighted Sobolev space.

Matrix weights and regularity for degenerate elliptic equations

We prove local boundedness, Harnack inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form. Degeneracy is via a non negative, symmetric, measurable matrix-valued function Q(x) and two suitable non negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev–Poincaré inequalities. Data integrability is close to L1 and it is exploited in terms of suitable version of Stummel-Kato class that in some cases is also necessary to the regularity.

Regularity results for a class of widely degenerate parabolic equations

Abstract Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE u t - div ⁡ ( ( | D ⁢ u | - ν ) + p - 1 ⁢ D ⁢ u | D ⁢ u | ) = f in ⁢ Ω T = Ω × ( 0 , T ) , u_{t}-\operatorname{div}\Bigl{(}(\lvert Du\rvert-\nu)_{+}^{p-1}\frac{Du}{% \lvert Du\rvert}\Bigr{)}=f\quad\text{in }\Omega_{T}=\Omega\times(0,T), where Ω is a bounded domain in ℝ n {\mathbb{R}^{n}} for n ≥ 2 {n\geq 2} , p ≥ 2 {p\geq 2} , ν is a positive constant and ( ⋅ ) + {(\,\cdot\,)_{+}} stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative u t {u_{t}} . The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius ν centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations [corrected version of mr2584740], J. Math. Pures Appl. (9) 93 2010, 6, 652–671], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations.

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries

This paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper, we proved an abstract result and discussed operators on the Heisenberg group. Here, we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and 0-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.

A De Giorgi Type Result to Divergence Degenerate Elliptic Equation with Bounded Coefficients Related To Hörmander's Vector Fields

A De Giorgi Type Result to Divergence Degenerate Elliptic Equation with Bounded Coefficients Related To Hörmander's Vector Fields

Existence and regularity results for a degenerate elliptic nonlinear equation on weighted Sobolev spaces

This paper deals with the existence of many solutions a degenerate weighted elliptic equation which its the number of solutions depend on degeneracy term a i.e., the number of subdomains of Ω\\a −1(0) whose boundary is made by submanifolds with 1-codimension. Moreover, by the Ljusternik-Schnirelman principle, we would study the corresponding eigenvalue problem and obtained the nonnegative eigenvalues sequence of main problem. Finally, we will discuss of the regularity of the solutions on C 1,α (Ω).

A Comparison Estimate for Singular p-Laplace Equations and Its Consequences

Comparison estimates are an important technical device in the study of regularity problems for quasilinear possibly degenerate elliptic and parabolic equations. Such tools have been employed indispensably in many papers of Mingione, Duzaar-Mingione, and Kuusi-Mingione, etc. on certain measure datum problems to obtain pointwise bounds for solutions and their full or fractional derivatives in terms of appropriate linear or nonlinear potentials. However, a comparison estimate for $p$-Laplace type elliptic equations with measure data is still unavailable in the strongly singular case $1< p\leq \frac{3n-2}{2n-1}$, where $n\geq 2$ is the dimension of the ambient space. This issue will be completely resolved in this work by proving a comparison estimate in a slightly larger range $1<p<3/2$. Applications include a `sublinear' Poincar\'e type inequality, pointwise bounds for solutions and their derivatives by Wolff's and Riesz's potentials, respectively. Some global pointwise and weighted estimates are also obtained for bounded domains, which enable us to treat a quasilinear Riccati type equation with possibly sublinear growth in the gradient.

Holder estimates and asymptotic behavior for degenerate elliptic equations in the half space

In this article we investigate the asymptotic behavior at infinity of viscosity solutions to degenerate elliptic equations. We obtain Holder estimates, up to the flat boundary, by using the rescaling method. Also as a byproduct we obtain a Liouville type result on Baouendi-Grushin type operators.

Fundamental solutions of a class of degenerate elliptic equation

In this paper, for the elliptic equation with two lines of different order of degeneration, given in the first quarter of the plane , four fundamental solutions are constructed that are expressed by Appell's hypergeometric functions of two variables. By means the decomposition formula of the Appell's function in products of the Gauss hypergeometric function, it is proved that the constructed fundamental solutions possess a logarithmic singularity for .

GRADIENT ESTIMATES FOR ENTROPY SOLUTIONS TO ELLIPTIC EQUATIONS WITH DEGENERATE COERCIVITY

We derive some a priori estimates and then prove the existence and regularity of distributional solutions to degenerate elliptic equations of the form {−div ⁡𝒜(x,u(x),∇ ⁡u(x))=f(x),x∈Ω,u(x)=0,x∈∂Ω, where the Carathéodory function 𝒜:Ω×ℝ×ℝn→ℝn satisfies 𝒜(x,s,ξ)⋅ξ≥α|ξ|p(1+|s|)𝜃,|𝒜(x,s,ξ)|≤β|ξ|p−1 for 1<p<n, 0≤𝜃<p−1 and 0<α≤β<∞, with f a Marcinkiewicz function. We obtain a gradient estimate for entropy solutions, and show that the result is optimal by a counterexample. We also consider degenerate elliptic equations with a lower order term, {−div ⁡𝒜(x,u(x),∇ ⁡u(x))+|u|r−1u=f(x),x∈Ω,u(x)=0,x∈∂Ω. We show that the presence of the lower order term has regularizing effects on our result.

Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations

We consider equations involving a combination of local and nonlocal degenerate p-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Hölder continuity with an explicit Hölder exponent in the general case. For certain parameters, our results also imply Hölder continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method.