The Principal Eigenvalue¶of a Conformally Invariant Differential Operator,¶with an Application to Semilinear Elliptic PDE

1999 ◽  
Vol 207 (1) ◽  
pp. 131-143 ◽  
Author(s):  
Matthew J. Gursky
Keyword(s):  
Differential Operator ◽  
Principal Eigenvalue ◽  
Elliptic Pde ◽  
Invariant Differential Operator ◽  
Conformally Invariant ◽  
Semilinear Elliptic ◽  
Invariant Differential

scholarly journals On an analytic description of the α-cosine transform on real Grassmannians

2016 ◽  
Vol 18 (02) ◽  
pp. 1550025 ◽  
Author(s):  
Semyon Alesker ◽  
Dmitry Gourevitch ◽  
Siddhartha Sahi
Keyword(s):  
Differential Operator ◽  
Radon Transform ◽  
Open Problem ◽  
Analytic Description ◽  
Radon Transforms ◽  
Invariant Differential Operator ◽  
Cosine Transform ◽  
Invariant Differential

The goal of this paper is to describe the [Formula: see text]-cosine transform on functions on real Grassmannian [Formula: see text] in analytic terms as explicitly as possible. We show that for all but finitely many complex [Formula: see text] the [Formula: see text]-cosine transform is a composition of the [Formula: see text]-cosine transform with an explicitly written (though complicated) [Formula: see text]-invariant differential operator. For all exceptional values of [Formula: see text] except one, we interpret the [Formula: see text]-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value [Formula: see text], which is [Formula: see text], is still an open problem.

scholarly journals Asymptotics for the Radon transform on hyperbolic spaces

2019 ◽  
pp. 241-251
Author(s):  
Nils Byrial ANDERSEN ◽  
Mogens FLENSTED-JENSEN
Keyword(s):  
Differential Operator ◽  
Maximal Compact Subgroup ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Hyperbolic Spaces ◽  
Schwartz Function ◽  
Invariant Differential Operator ◽  
Abel Transform ◽  
Invariant Functions ◽  
Invariant Differential

Let G/H be a hyperbolic space over R; C or H; and let K be a maximal compact subgroup of G. Let D denote a certain explicit invariant differential operator, such that the non-cuspidal discrete series belong to the kernel of D. For any L^2-Schwartz function f on G/H we prove that the Abel transform A(Df) of Df is a Schwartz function. This is an extension of a result established in [2] for K-finite and K∩H-invariant functions.

scholarly journals HOMOGENIZATION OF A PERIODIC DEGENERATE SEMILINEAR ELLIPTIC PDE

2011 ◽  
Vol 11 (02n03) ◽  
pp. 475-493 ◽  
Author(s):  
ÉTIENNE PARDOUX ◽  
AHMADOU BAMBA SOW
Keyword(s):  
Differential Operator ◽  
Weak Convergence ◽  
Diffusion Processes ◽  
Probabilistic Method ◽  
Second Order ◽  
Elliptic Pde ◽  
Semilinear Elliptic ◽  
Terminal Time ◽  
Oscillating Coefficients ◽  
Order Part

In this paper, a semilinear elliptic PDE with rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of ℝd. Our fully probabilistic method is based on the connection between PDEs and BSDEs with random terminal time and the weak convergence of a class of diffusion processes.

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