Browsing by Author "Spector, Daniel E."

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On a new class of fractional partial differential equations
(2015-10-01)
Shieh, Tien-Tsan
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Spector, Daniel E.
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應用數學系
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Department of Applied Mathematics
In this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.
On a new class of fractional partial differential equations II
(2018-07-01)
Shieh, Tien-Tsan
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Spector, Daniel E.
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應用數學系
;
Department of Applied Mathematics
In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular, we here establish an L-1 Hardy inequality, obtain further regularity results for solutions of certain fractional PDE, demonstrate the existence of minimizers for integral functionals of the fractional gradient with non-linear dependence in the field, and also establish the existence of solutions to corresponding Euler-Lagrange equations obtained as conditions of minimality. In addition, we pose a number of open problems, the answers to which would fill in some gaps in the theory as well as to establish connections with more classical areas of study, including interpolation and the theory of Dirichlet forms.
Regularity for a fractional p-Laplace equation
(2018-02-01)
Schikorra, Armin
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Shieh, Tien-Tsan
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Spector, Daniel E.
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應用數學系
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Department of Applied Mathematics
In this note, we consider regularity theory for a fractional p-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the H-s,H-p-Laplacian. We obtain the natural analogue to the classical p-Laplacian situation, namely C-loc(s+alpha)-regularity for the homogeneous equation.
Uniqueness of Equilibrium with Sufficiently Small Strains in Finite Elasticity
(2019-07-01)
Spector, Daniel E.
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Spector, Scott J.
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應用數學系
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Department of Applied Mathematics
The uniqueness of equilibrium for a compressible, hyperelastic body subject to dead-load boundary conditions is considered. It is shown, for both the displacement and mixed problems, that there cannot be two solutions of the equilibrium equations of Finite (Nonlinear) Elasticity whose nonlinear strains are uniformly close to each other. This result is analogous to the result of JOHN (Commun Pure Appl Math 25:617-634, 1972), who proved that, for the displacement problem, there is a most one equilibrium solution with uniformly small strains. The proof in this manuscript utilizes Geometric Rigidity, a new straightforward extension of the Fefferman-Stein inequality to bounded domains, and an appropriate adaptation, for Elasticity, of a result from the Calculus of Variations. Specifically, it is herein shown that the uniform positivity of the second variation of the energy at an equilibrium solution implies that this mapping is a local minimizer of the energy among deformations whose gradient is sufficiently close, in BMO boolean AND L-1 , to the gradient of the equilibrium solution.