Holder Estimates for Higher Dimensional Corner Models

Author(s):

Charles L. Epstein ◽

Rafe Mazzeo

Keyword(s):

Cauchy Problem ◽

Laplace Transform ◽

Heat Kernel ◽

Resolvent Operator ◽

Higher Dimensions ◽

Euclidean Case ◽

Higher Dimensional ◽

The Laplace Transform ◽

The Resolvent Operator ◽

This chapter establishes Hölder space estimates for higher dimensional corner model problems. It first explains the homogeneous Cauchy problem before estimating the solution of the inhomogeneous problem in a n-dimensional corner. It then reduces the proof of an estimate in higher dimensions to the estimation of a product of 1-dimensional integrals. Using the “1-variable-at-a-time” method, the chapter proves the higher dimensional estimates in several stages by considering the “pure corner” case where m = 0, and then turns to the Euclidean case, where n = 0. It also discusses the resolvent operator as the Laplace transform of the heat kernel.

The Resolvent Operator

10.23943/princeton/9780691157122.003.0011 ◽

2017 ◽

Author(s):

Charles L. Epstein ◽

Rafe Mazzeo

Keyword(s):

Cauchy Problem ◽

Heat Equation ◽

Laplace Transform ◽

Heat Kernel ◽

Resolvent Operator ◽

Contour Integration ◽

Resolvent Kernel ◽

The Laplace Transform ◽

Elliptic Estimates ◽

The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

Holder Estimates for Euclidean Models

10.23943/princeton/9780691157122.003.0008 ◽

2017 ◽

Author(s):

Charles L. Epstein ◽

Rafe Mazzeo

Keyword(s):

Cauchy Problem ◽

Heat Kernel ◽

Resolvent Operator ◽

Second Derivative ◽

Kernel Estimates ◽

First Derivative ◽

Essential Components ◽

The Laplace Transform ◽

The Resolvent Operator ◽

This chapter presents the Hölder space estimates for Euclidean model problems. It first considers the homogeneous Cauchy problem and the inhomogeneous problem before defining the resolvent operator as the Laplace transform of the heat kernel. It then describes the 1-dimensional kernel estimates that form essential components of the proofs of the Hölder estimates for the general model problems; these include basic kernel estimates, first derivative estimates, and second derivative estimates. The proofs of these estimates are elementary. The chapter concludes by proving estimates on the resolvent and investigating the off-diagonal behavior of the heat kernel in many variables.

Holder Estimates for the 1-dimensional Model Problems

10.23943/princeton/9780691157122.003.0006 ◽

2017 ◽

Author(s):

Charles L. Epstein ◽

Rafe Mazzeo

Keyword(s):

Heat Kernel ◽

Resolvent Operator ◽

Heat Kernels ◽

Dimensional Model ◽

Kernel Estimates ◽

Schauder Estimates ◽

Diffusion Operator ◽

Higher Dimensional ◽

The Resolvent Operator ◽

This chapter establishes Hölder space estimates for the 1-dimensional model problems. It gives a detailed treatment of the 1-dimensional case, in part because all of the higher dimensional estimates are reduced to estimates on heat kernels for the 1-dimensional model problems. It also presents the proof of parabolic Schauder estimates for the generalized Kimura diffusion operator using the explicit formula for the heat kernel, along with standard tools of analysis. Finally, it considers kernel estimates for degenerate model problems, explains how Hölder estimates are obtained for the 1-dimensional model problems, and describes the properties of the resolvent operator.

Holder Estimates for General Models

10.23943/princeton/9780691157122.003.0009 ◽

2017 ◽

Author(s):

Charles L. Epstein ◽

Rafe Mazzeo

Keyword(s):

Cauchy Problem ◽

General Model ◽

Resolvent Operator ◽

Heat Equations ◽

Second Derivative ◽

Time Behavior ◽

Long Time Behavior ◽

Long Time ◽

The Resolvent Operator ◽

This chapter presents the Hölder estimates for general model problems. It first estimates solutions to heat equations for both the homogeneous Cauchy problem and the inhomogeneous problem, obtaining first and second derivative estimates in the latter case, before discussing a general result describing the off-diagonal and long-time behavior of the solution kernel for the general model. It also states a proposition summarizing the properties of the resolvent operator as an operator on the Hölder spaces. In contrast to the case of the heat equation, there is no need to assume that the data has compact support in the x-variables to prove estimates when k > 0.

The extinction time of a general birth and death process with catastrophes

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Author(s):

P. J. Brockwell

Keyword(s):

Laplace Transform ◽

Size Distribution ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Death Process ◽

Extinction Time ◽

Birth And Death Process ◽

Different Types ◽

The Laplace Transform ◽

Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Solution of a Mathematical Model Describing the Change of Hormone Level in Thyroid Using the Laplace Transform

Mathematical Journal of Interdisciplinary Sciences ◽

10.15415/mjis.2013.21009 ◽

2013 ◽

Vol 2 (1) ◽

pp. 99-108

Author(s):

Shama M. Mulla ◽

Chhaya H. Desai

Keyword(s):

Mathematical Model ◽

Laplace Transform ◽

Hormone Level ◽

The Laplace Transform

Controllability for some nonlinear impulsive partial functional integrodifferential systems with infinite delay in Banach spaces

Open Journal of Mathematical Analysis ◽

10.30538/psrp-oma2020.0069 ◽

2020 ◽

Vol 4 (2) ◽

pp. 104-115

Author(s):

Khalil Ezzinbi ◽

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Banach Spaces ◽

Integrodifferential Equation ◽

Resolvent Operator ◽

Measure Of Noncompactness ◽

Infinite Delay ◽

Sufficient Conditions ◽

Mönch Fixed Point Theorem ◽

The Resolvent Operator

This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of the work of K. Balachandran and R. Sakthivel (Journal of Mathematical Analysis and Applications, 255, 447-457, (2001)) and a host of important results in the literature, without assuming the compactness of the resolvent operator. An example is given for illustration.