Nonlinear Potential Theory Research Articles

Nonlinear potential theory and PDEs

We consider equations like -div(|∇u|p−2∇u)=μ, where μ is a nonnegative Radon measure and 1<p<∞. Results that relate the solutionu and the measure μ are reviewed. A link between potential estimates and the boundary regularity of the Dirichlet problem is established.

Classification of Riemannian manifolds in nonlinear potential theory

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.

Existence of dirichlet finite harmonic measures in nonlinear potential theory

It is shown that, although there exist no nonconstant 2-Dirichlet finite 2-harmonic measures on the unit disc, there exist nonconstant p-Dirichlet finite p-harmonic measures on the unit disc for every p with 1<p<2.

Solution of a problem of nonlinear potential theory by the fixed domain method

A solution is derived for a one-dimensional boundary-value problem of nonlinear potential theory with one free end with a supplementary boundary condition. The problem is solved by a variant of the fixed domain method. In each fixed domain, the problem is reduced to the corresponding nonlinear difference problem, which is solved by Newton's method.

Weighted Lp-potential theory on homogeneous groups

Two independent generalizations of nonlinear potential theory [1-3] are combined. The first generalization [4] was made in response to boundary-value problems for degenerate elliptical equations [5, 6], and the second [7-10] was motivated by certain problems related to anisotropic spaces of differentiable functions [11-16] and investigations in the area of quasielliptical equations [17, 18]. The approach to nonlinear potential theory proposed in the present article comprises two generalizations, and a number of propositions stated here for weighted potential theory are novel, for example, Frostman-type theorems concerning the relation between the weighted capacity and weighted Hausdorff volume (measure). Strong Maz'ya-type capacity inequalities [19] and two-weighted bounds for general type potentials are introduced as applications of the theory in the general situation. In conclusion weight-theoretic analogues of the Choquet theorem and Kellog properties are formulated and applications of these analogues to degenerate elliptical equations are presented.

Harmonic morphisms in nonlinear potential theory

This article concerns the following problem: given a family of partial differential operators with similar structure and given a continuous mapping f from an open set Ω in Rn into Rn, then when does f pull back the solutions of one equation in the family to solutions of another equation in that family? This problem is typical in the theory of differential equations when one wants to use a coordinate change to study solutions in a different environment.

Harmonic morphisms and non-linear potential theory

Harmonic morphisms and non-linear potential theory

Best uniform approximation by 𝔎-superharmonic functions

Let &#x1d50e; be a sheaf of continuous real-valued functions on a domain . If &#x1d50e; satisfies the axioms for a nonlinear potential theory, then we call the elements of &#x1d50e;(U) for U ⊂Ω open&#x1d50e;-harmonic on U.We define the class &#x1d516;(U) of &#x1d50e;-superharmonic functions on Uand characterize best uniform approximations to continuous functions by elements of &#x1d516;(U) ∩ C([Ubar]).

Generalized dirichlet problem in nonlinear potential theory

The operator extending the classical solution of the Dirichlet problem for the quasilinear elliptic equation divA(x,▽u)=0 akin to thep-Laplace equation is shown to be unique providedA obeys a specific order principle. The Keldych lemma is also generalized to this nonlinear setting.

A BOUNDARY ELEMENT ANALYSIS FOR NONLINEAR-WAVE DIFFRACTION AROUND LARGE STRUCTURE

A boundary element method using nonlinear potential theory has been developed for analyzing wave diffraction around large structure. To confirm the validity of the proposed method, the numerical results for solitary wave propagating through a three-dimensional wave channel are compared with analytical solutions in terms of wave profile and hydrodynamic pressure. The comparison shows excellent agreement regardless of H0/h (H0: wave height, h: water depth). Furthermore, nonlinear effects on wave loads exerted by solitary wave on a vertical cylinder are investigated by comparing the numerical results with first approximations given by Isaacson. The first approximations for horizontal force, overturning moment and wave run-up are appropriate with H0/h being 0.1. It is, however, found that the first approximations for overturning moment are underestimated remarkably with H0/h being 0.4.

A BOUNDARY ELEMENT ANALYSIS FOR CNOIDAL WAVE RUN-UP

A boundary element method using nonlinear potential theory has been developed for analyzing cnoidal wave run-up on a plane with a constant inclination angle. The validity of the method is confirmed by comparing the numerical results with experimental data. Because of the wave nonlinear effect, the linear solution given by Shuto predicts a much smaller magnitude of wave run-up. In addition, the applicability of Takada's formula, which takes the wave nonlinearity into consideration, is investigated by comparing the results obtained from the formula with those of the boundary element analysis. It is found that Takada's formula does not yield a good approximation for the run-up height of a cnoidal wave.

Weighted nonlinear potential theory

The potential theoretic idea of the "thinness of a set at a given point" is extended to the weighted nonlinear potential theoretic setting—the weights representing in general singularities/degeneracies—and conditions on these weights are given that guarantee when two such notions are equivalent at the given point. When applied to questions of boundary regularity for solutions to (degenerate) elliptic second-order partial differential equations in bounded domains, this result relates the boundary Wiener criterion for one operator to that of another, and in the linear case gives conditions for boundary regular points to be the same for various operators. The methods also yield two weight norm inequalities for Riesz potentials \[ ( ∫ ( I α ∗ f ) q v d x ) 1 / q ⩽ ( ∫ f p w d x ) 1 / p , {\left ( {\int {{{({I_\alpha }{\ast }f)}^q}v\,dx} } \right )^{1/q}} \leqslant {\left ( {\int {{f^p}w\,dx} } \right )^{1/p}}, \] 1 &gt; p ⩽ q &gt; ∞ 1 &gt; p \leqslant q &gt; \infty , which at least in the first-order case ( α = 1 ) (\alpha = 1) have found some use in a number of places in analysis.

Fine and quasi connectedness in nonlinear potential theory

Si B α,p désigne la capacité de Bessel des sous-ensembles de l’espace euclicien de dimension n, α>0, 1<p<∞, associé naturellement avec l’espace des potentiels de Bessel des fonctions L p -functions, alors notre résultat principal est l’estimation suivante : pour 1<αp≤n, il existe une constante C=C(α,p,n) de telle sorte que pour n’importe quel ensemble E,

Strip Basin Recharge to Inclined Water Table

The hydrodynamics of recharge mounds that develop on an inclined water table beneath a strip recharge basin are considered. The nonlinear potential (NLP) theory is used to predict the motion of the water table resulting from the recharge operation. The theoretical derivation of the NLP theory uses the saturated flow equations for a homogeneous isotropic medium. Solutions to the NLP theory for recharge to an inclined water table are obtained by using a finite difference method. A few examples of flow patterns showing the motion of the recharged and native water are given. The mathematical model can be utilized to predict the response and movement of recharged water in the design and operation of recharge facilities.

Thin sets in nonlinear potential theory

Thin sets in nonlinear potential theory

On the Edge Singularity of an Actuator Disk with Large Constant Normal Load

In this paper some aspects of the nonlinear potential theory of actuator disks are considered. A rather general formulation of the problem for a prescribed load on a curved surface is given. For the special case of constant normal load and no incoming velocity the singular behavior of the flow at the edge of the disk is discussed.

Dupuit‐Forchheimer and potential theories for recharge from basins

Artificial groundwater recharge from strip and square basins to an unconfined aquifer was investigated. The classical Dupuit‐Forchheimer (DF) theory and the more exact potential theory were used to determine the motion of the water table resulting from recharge. For each theory, both a linear and a nonlinear version were considered. Finite difference schemes were employed to solve the nonlinear models. Analytical solutions exist for the linear models. By comparing the results from the various theories and from experimental results in a sand model the range of validity of the theories was determined. For sufficiently shallow flows the DF theories give results that are good approximations to the potential theories. For sufficiently deep flows the potential theories must be used, particularly beneath the basin and for small times. The velocity field near a strip basin was determined from the nonlinear potential theory for both a deep and a shallow flow. Tabular results are presented that can be used to determine the rise of the water table for strip and square basins.

The dirichlet problem for a complex Monge-Amp�re equation

where G' denotes the Jacobian determinant of G. Furthermore, if G4=(0 . . . . . 0), then (d d ~ log h G I) 0. Thus, for f = 0, (1) is a natural generalization of the Dirichlet problem for harmonic in the complex plane. Other extended Dirichlet problems were studied in connection with function theory in several variables by S. Bergman [2, 3] (on domains with distinguished boundary surfaces) and more generally by H. Bremermann [4]. In Section 8 it is shown that the solution of the problem discussed by Bremermann actually solves (1), in a generalized sense, with f = 0 . The problem (1) seems to be a reasonable candidate for a (nonlinear) potential theory associated with the theory of of several complex variables. The question of uniqueness for the problem (1) is related to the question of existence of inner functions on the domain O. If h is a bounded analytic function

The Dirichlet problem for a complex Monge-Ampere equation

where G' denotes the Jacobian determinant of G. Furthermore, if G4=(0 . . . . . 0), then (d d ~ log h G I) 0. Thus, for f = 0, (1) is a natural generalization of the Dirichlet problem for harmonic in the complex plane. Other extended Dirichlet problems were studied in connection with function theory in several variables by S. Bergman [2, 3] (on domains with distinguished boundary surfaces) and more generally by H. Bremermann [4]. In Section 8 it is shown that the solution of the problem discussed by Bremermann actually solves (1), in a generalized sense, with f = 0 . The problem (1) seems to be a reasonable candidate for a (nonlinear) potential theory associated with the theory of of several complex variables. The question of uniqueness for the problem (1) is related to the question of existence of inner functions on the domain O. If h is a bounded analytic function