Gradient continuity estimates for the normalized p-Poisson equation

In [1, 2, 3, 4, 6], the authors posed and discussed some boundary‐value problems of second order elliptic equations with parabolic degeneracy. In [5], the authors posed and discussed some boundary‐value problems of second order mixed equations with degenerate curve on the sides of an angle, but the coefficients of the equations possess strong restrictions. The present paper deals with the mixed problem for elliptic equations with degenerate curve on the sides of an angle, where the coefficients satisfy general conditions. We first give the formulation of the problem and estimates of solutions of the problem for the equations, and then prove the existence of solutions for the above problem by the Leray–Schauder theorem. In this paper, we use the complex analytic method, namely we first introduce the new notation (2.1) below and reduce the degenerate elliptic equations of second order to the corresponding problem for degenerate elliptic complex equations of first order, afterwards the above problem of second order degenerate elliptic equations can be solved. The results in this paper will be used in a subsequent paper to handle the Tricomi problem of second order equations of mixed type with degenerate curve on the sides of an angle. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Compactness Methods for Certain Degenerate Elliptic Equations

Structures in circumstellar matter reflect both fast processes and quasi-equilibrium states. A geometrical diversity of emitting circumstellar matter is observed around evolved massive stars, in particular around B[e] supergiants. We recapitulate classical analytical tools of linear and nonlinear potential theory, such as Cole–Hopf transformation and Grad–Shafranov theory, and develop them further to explain the occurrence of the circumstellar matter structures and their dynamics. We use potential theory to formulate the nonlinear hydrodynamical equations and test dilatations of the quasi-equilibrium initial conditions. We find that a wide range of flow patterns can basically be generated and the timescales can switch, based on initial conditions, and lead to eruptive processes, reinforcing that the nonlinear fluid environment includes both quasi-stationary structures and fast processes like finite-time singularities. Some constraints and imposed symmetries can lead to Keplerian orbits, while other constraints can deliver quasi-Keplerian ones. The threshold is given by a characteristic density at the stellar surface.

The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations

The classification theory of Riemann surfaces is generalized to Riemanniann-manifolds in the conformally invariant case. This leads to the study of the existence ofA-harmonic functions of typen with various properties and to an extension of the definition of the classical notions with inclusionsOG⊂OHP⊂OHB⊂OHD. In the classical case the properness of the inclusions were proved rather late, in the 50's by Ahlfors and Toki. Our main objective is to show that such inclusions are proper also in the generalized case.

Originally, harmonic morphisms were defined as continuous mappings φ:X → X' between harmonic spaces such that h'∘φ remains harmonic whenever h' is harmonic, see [1], p. 20. In general linear axiomatic potential theory, one has to replace harmonic function

Approximation in Sobolev spaces and nonlinear potential theory

Pointwise gradient bounds via Riesz potentials, such as those available for the linear Poisson equation, actually hold for general quasilinear degenerate equations of p-Laplacean type. The regularity theory of such equations completely reduces to that of the classical Poisson equation up to the C1-level.

<div class="htmlview paragraph">This paper describes a software system called FREPS (Forced REsponse Prediction System) that integrates structural dynamic, steady and unsteady aerodynamic analyses to efficiently predict the forced response dynamic stresses in axial flow turbomachinery blades due to aerodynamic and mechanical excitations. A flutter analysis capability is also incorporated into the system. The FREPS system performs aeroelastic analysis by modeling the motion of the blade in terms of its normal modes. The structural dynamic analysis is performed by a finite element code such as MSC/NASTRAN. The steady aerodynamic analysis is based on nonlinear potential theory and the unsteady aerodynamic analysis is based on a linearization of the non-uniform potential mean flow. The program description and presentation of the capabilities are reported herein. The effectiveness of the FREPS package is demonstrated on the High Pressure Oxygen Turbopump turbine of the Space Shuttle Main Engine. Both flutter and forced response analyses are performed and typical results are illustrated.</div>

Nonlinear potential theory in metric spaces

We consider a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions. Existence of solutions of Dirichlet problem for equations of the given class in the case where these equations have L 1-right-hand sides depending on unknown function has been established in our recent article. In the present article we prove some results which show the improvement of summability of solutions of the same problem under additional conditions on right-hand sides of the equations under consideration.

We study weak geodesics in the space of potentials for the deformed Hermitian–Yang–Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey–Lawson. By exploiting the convexity of the level sets of the Lagrangian angle operator in the highest branch, we are able to construct $C^0$ solutions of the associated Dirichlet problem.

RETRACTED: Quartic spline collocation methods for elliptic partial differential equations

OF THE DISSERTATION On some nonlocal elliptic and parabolic equations by Tianling Jin Dissertation Director: YanYan Li We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem involving nonlocal conformally invariant operators. Regularity properties for solutions of some degenerate elliptic equations as well as a Liouville type theorem are established, and used in our blow up analysis. We also introduce a fractional Yamabe flow and show that on the conformal spheres (Sn, [gSn ]) it converges to the standard sphere up to a Mobius diffeomorphism. These arguments can be applied to obtain extinction profiles of solutions of some fractional porous medium equations, which are further used to improve a Sobolev inequality via a quantitative estimate of the remainder term.

The mixed problem for a nonlinear degenerate elliptic equation of second order