Existence and uniqueness of the solution of linear degenerate PDEs in weighted Sobolev space with nonhomogeneous boundary condition

We prove the existence and uniqueness of solution to the nonhomogeneous degenerate elliptic PDE of second order with boundary data in weighted Orlicz‐Slobodetskii space. Our goal is to consider the possibly general assumptions on the involved constraints: the class of weights, the boundary data, and the admitted coefficients. We also provide some estimates on the spectrum of our degenerate elliptic operator.     

Analytic approach to solve a degenerate parabolic PDE for the Heston model

We present an analytic approach to solve a degenerate parabolic problem associated with the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second‐order degenerate elliptic operator of the parabolic PDE. We use this approach to prove, under appropriate assumptions on some involved unknown parameters, the existence and uniqueness of weak solutions to the parabolic problem on unbounded subdomains of the half‐plane. Copyright © 2017 John Wiley & Sons, Ltd.     

Semilinear Elliptic Equations on Thin Network-Shaped Domains with Variable Thickness

We study semilinear elliptic equations in a thin domain which is shaped like a network and degenerate into a geometric graph when a certain parameter tends to zero. It is shown by using a comparison technique that a solution of the equation in the network-shaped domain approaches a solution of an associated limit equation on the graph. Conversely, when the limit equation on the graph has a solution, we prove the existence of a solution of the PDE in the network-shaped domain which converges to the solution of the limit equation.     

Discrete and continuous maximum principles for parabolic and elliptic operators

The major qualitative properties of linear parabolic and elliptic operators/PDEs are the different maximum principles (MPs). Another important property is the stabilization property (SP), which connects these two types of operators/PDEs. This means that under some assumptions the solution of the parabolic PDE tends to an equilibrium state when t→∞, which is the solution of the corresponding elliptic PDE. To solve PDEs we need to use some numerical methods, and it is a natural requirement that these qualitative properties are preserved on the discrete level. In this work we investigate this question when a two-level discrete mesh operator is used as the discrete model of the parabolic operator (which is a one-step numerical procedure for solving the parabolic PDE) and a matrix as a discrete elliptic operator (which is a linear algebraic system of equations for solving the elliptic PDE). We clarify the relation between the discrete parabolic maximum principle (DPMP), the discrete elliptic maximum principle (DEMP) and the discrete stabilization property (DSP). The main result is that the DPMP implies the DSP and the DEMP.     

Krylov and Safonov estimates for degenerate quasilinear elliptic PDEs

We here establish an a priori Hölder estimate of Krylov and Safonov type for the viscosity solutions of a degenerate quasilinear elliptic PDE of non-divergence form. The diffusion matrix may degenerate when the norm of the gradient of the solution is small: the exhibited Hölder exponent and Hölder constant only depend on the growth of the source term and on the bounds of the spectrum of the diffusion matrix for large values of the gradient. In particular, the given estimate is independent of the regularity of the coefficients. As in the original paper by Krylov and Safonov, the proof relies on a probabilistic interpretation of the equation.     

Sequential and continuum bifurcations in degenerate elliptic equations

We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations.

     

A New Elliptic Measure on Lower Dimensional Sets

The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n-1 dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.     

Some new applications of the method of lower and upper solutions to elliptic problems

Motivated by the application to some degenerate elliptic problems (here, degenerate means nonlinear diffusion), we propose a new monotone scheme of iteration which provides the existence of a minimal and a maximal solution between a sub-solution and a (greater) super-solution. We apply this result to some problems arising from biology.     

A Caffarelli–Kohn–Nirenberg inequality in Orlicz–Sobolev spaces and applications

A generalization of the classical Caffarelli–Kohn–Nirenberg inequality is obtained in the setting of Orlicz–Sobolev spaces. As applications, we prove a compact embedding result, and we establish the existence of weak solutions of the Dirichlet problem for a nonhomogeneous and degenerate/singular elliptic PDE.     

Elliptic equations strongly degenerate at a point

In this paper we will mainly propose some problems for a class of degenerate elliptic equations, either linear or nonlinear. We will study some special cases of these problems and reveal some phenomena which may not have been noticed previously.     

Remarks on non homogeneous elliptic Kirchhoff equations

Extending the results in (Ambrosetti and Arcoya, Positive solutions of elliptic Kirchhoff equations (to appear)), the existence of solutions for some classes of nonlinear elliptic, possibly degenerate, non homogeneous Kirchhoff equations are proved. We handle smooth nonlinearities as well as the case in which the nonlinearity has an upward jump discontinuity.     

The mixed problem for degenerate elliptic equations of second order

The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order.     

Nonlinear transform and Jacobi elliptic function solutions of nonlinear equations

The nondegenerative elliptic function solutions of some nonlinear equations are obtained by a nonlinear transform, which names the Jacobi elliptic function expansion. When taking particular parameters, the elliptic function solutions can degenerate as solitary wave solutions and singularity solutions.     

Existence results for a class of degenerate quasilinear elliptic systems

We establish existence results for weak solutions of degenerate quasilinear elliptic systems. By using the variational method we obtain the existence of a solution for an elliptic system with Dirichlet boundary condition under some restriction on λ.     

A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.     

On very weak solutions of degenerate p-harmonic equations

We establish a regularity result for very weak solutions of some degenerate elliptic PDEs. The nonnegative function which measures the degree of degeneracy of ellipticity bounds is assumed to be exponentially integrable. We find that the scale of improved regularity is logarithmic.      

A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time‐dependent ODE modeling the evolution of the gating variable. In the simpler subcase of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge‐Kutta‐Fehlberg‐type adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding nonsignificant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed‐up, memory compression, and errors in different norms. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

On the continuity of minimizers for quasilinear functionals

In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable. The minimizers are shown to have a modulus of continuity controlled by log log(1/|x|)⁻¹. Our proof adapts ideas developed for solutions of degenerate elliptic equations by J. Onninen, X. Zhong: Continuity of solutions of linear, degenerate elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 103–116.     

Regularizing effects of some lower order terms in non-uniformly nonlinear elliptic equations

We consider the Dirichlet problem for a class of strongly nonlinear elliptic equations with degenerate coercivity and data in divergence form. We show that some lower order terms have regularizing effects on solutions.     

Local Solutions of Weakly Parabolic Quasilinear Differential Equations

We consider a quasilinear parabolic boundary value problem, the elliptic part of which degenerates near the boundary. In order to solve this problem, we approximate it by a system of linear degenerate elliptic boundary value problems by means of semidiscretization with respect to time. We use the theory of degenerate elliptic operators and weighted Sobolev spaces to find a priori estimates for the solutions of the approximating problems. These solutions converge to a local solution, if the step size of the time-discretization goes to zero. It is worth pointing out that we do not require any growth conditions on the nonlinear coefficients and right-hand side, since we lire able to prove L∞ - estimates.