Maximum principle in potential theory and imbedding theorems for anisotropic spaces of differentiable functions

1988 ◽  
Vol 29 (2) ◽  
pp. 176-189 ◽  
Author(s):  
S. K. Vodop'yanov
Keyword(s):  
Maximum Principle ◽  
Potential Theory ◽  
Differentiable Functions ◽  
Anisotropic Spaces

scholarly journals Potential theory for manifolds with boundary and applications to controlled mean curvature graphs

2017 ◽  
Vol 2017 (733) ◽  
Author(s):  
Debora Impera ◽  
Stefano Pigola ◽  
Alberto G. Setti
Keyword(s):  
Maximum Principle ◽  
Potential Theory ◽  
Mean Curvature ◽  
Manifolds With Boundary ◽  
Riemannian Product ◽  
Prescribed Mean Curvature ◽  
New Information ◽  
New Form

AbstractIn this paper we characterize the Neumann-parabolicity of manifolds with boundary in terms of a new form of the classical Ahlfors maximum principle and of a version of the so-called Kelvin–Nevanlinna–Royden criterion. The motivation underlying this study is to obtain new information on the geometry of graphs with prescribed mean curvature inside a Riemannian product of the type

scholarly journals On a maximum principle in potential theory

1982 ◽  
Vol 107 (4) ◽  
pp. 346-359
Author(s):  
Dagmar Křivánková
Keyword(s):  
Maximum Principle ◽  
Potential Theory

scholarly journals An Example of a Positive Non-Symmetric Kernel Satisfying the Complete Maximum Principle

1972 ◽  
Vol 48 ◽  
pp. 189-196 ◽  
Author(s):  
Masanori Kishi
Keyword(s):  
Maximum Principle ◽  
Potential Theory ◽  
Real Line ◽  
The Real ◽  
Symmetric Kernel ◽  
Symmetric Convolution ◽  
Convolution Kernels ◽  
Complete Maximum Principle

One of the main problems in potential theory is to determine the class of kernels satisfying the domination principle or the complete maximum principle. As to positive symmetric kernels this is settled to a certain extent, but as to non-symmetric kernels we have not yet obtain satisfactorily large explicit classes. In this note we shall give a class of positive non-symmetric convolution kernels on the real line satisfying the complete maximum principle.

scholarly journals Maximum Principles in the Potential Theory

1963 ◽  
Vol 23 ◽  
pp. 165-187 ◽  
Author(s):  
Masanori Kishi
Keyword(s):  
Maximum Principle ◽  
Potential Theory ◽  
Positive Measure ◽  
Hausdorff Space ◽  
Finite Energy ◽  
Maximum Principles ◽  
Compact Set ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Locally Compact Hausdorff Space

Ninomiya, in his thesis [13] on the potential theory with respect to a positive symmetric continuous kernel G on a locally compact Hausdorff space Ω, proves that G satisfies the balayage (resp. equilibrium) principle if and only if G satisfies the domination (resp. maximum) principle. He starts from the Gauss-Ninomiya variation and shows that for any given compact set K in Ω and any positive upper semi-continuous function u on K, there exists a positive measure μ on K such that its potential Gμ is ≥ u on the support of μ and Gμ≥u on K almost everywhere with respect to any positive measure with finite energy.

scholarly journals Sario’s Potentials and Analytic Mappings

1967 ◽  
Vol 29 ◽  
pp. 93-101
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Maximum Principle ◽  
Potential Theory ◽  
Fundamental Theorem ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Value Distribution Theory ◽  
Complete Form ◽  
Fundamental Theorems ◽  
Analytic Mappings

In order to extend Nevanlinna’s first and second fundamental theorems to arbitrary analytic mappings between Riemann surfaces, Sario [8, 9] introduced a kernel function on an arbitrary Riemann surface generalizing the elliptic kernel on the Riemann sphere. Because of the importance of the potential theoretic method in the value distribution theory, we discussed potentials of Sario’s kernel in [4]. In that paper the validty of Frostman’s maximum principle for Sario’s potentials was left unsettled. The main object of this paper is to resolve this question (Theorem 1). As a consequence the fundamental theorem of the potential theory is obtained in its complete form for Sario’s potentials (Theorem 2).

scholarly journals The embedding theorems for anisotropic Nikol’skii-Besov spaces with generalized mixed smoothness

2021 ◽  
Vol 104 (4) ◽  
pp. 28-34
Author(s):  
K.A. Bekmaganbetov ◽  
◽  
K.Ye. Kervenev ◽  
Ye. Toleugazy ◽  
◽  
...  
Keyword(s):  
Approximation Theory ◽  
Mathematical Physics ◽  
Trigonometric Polynomials ◽  
Embedding Theorems ◽  
Lebesgue Spaces ◽  
Differentiable Functions ◽  
Generalized Mixed Smoothness ◽  
Anisotropic Spaces ◽  
Mixed Metric ◽  
Mixed Smoothness

The theory of embedding of spaces of differentiable functions studies the important relations of differential (smoothness) properties of functions in various metrics and has a wide application in the theory of boundary value problems of mathematical physics, approximation theory, and other fields of mathematics. In this article, we prove the embedding theorems for anisotropic spaces Nikol’skii-Besov with a generalized mixed smoothness and mixed metric, and anisotropic Lorentz spaces. The proofs of the obtained results are based on the inequality of different metrics for trigonometric polynomials in Lebesgue spaces with mixed metrics and interpolation properties of the corresponding spaces.

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