scholarly journals MINIMIZERS AND GAMMA-CONVERGENCE OF ENERGY FUNCTIONALS DERIVED FROM $p$-LAPLACIAN EQUATION

2009 ◽  
Vol 13 (6B) ◽  
pp. 2021-2036 ◽  
Author(s):  
Mao-Sheng Chang ◽  
Shu-Cheng Lee ◽  
Chien-Chang Yen
Keyword(s):  
Gamma Convergence ◽  
Energy Functionals ◽  
Laplacian Equation

scholarly journals The Systems of Nonlinear Gradient Flows on Metric Spaces and Its Gamma-Convergence

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Mao-Sheng Chang ◽  
Bo-Cheng Lu
Keyword(s):  
Nonlinear System ◽  
Metric Spaces ◽  
Gradient Flow ◽  
Gamma Convergence ◽  
Gradient Flows ◽  
Energy Functionals ◽  
Several Variables ◽  
Flow Systems

We first establish the explicit structure of nonlinear gradient flow systems on metric spaces and then develop Gamma-convergence of the systems of nonlinear gradient flows, which is a scheme meant to ensure that if a family of energy functionals of several variables depending on a parameter Gamma-converges, then the solutions to the associated systems of gradient flows converge as well. This scheme is a nonlinear system edition of the notion initiated by Sylvia Serfaty in 2011.

scholarly journals Geometric linearization of theories for incompressible elastic materials and applications

2021 ◽  
pp. 1-32
Author(s):  
Martin Jesenko ◽  
Bernd Schmidt
Keyword(s):  
Small Displacement ◽  
Gamma Convergence ◽  
Stored Energy ◽  
Energy Functionals ◽  
Small Length Scale ◽  
Nematic Elastomers ◽  
Incompressible Materials ◽  
Geometric Linearization ◽  
Displacement Regime ◽  
Nonlinear Energy

We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.

scholarly journals $L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hui Wang ◽  
Caisheng Chen
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Divergence Form ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Estimates Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

scholarly journals Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

scholarly journals Regularity and time behavior of the solutions to weak monotone parabolic equations

2021 ◽  
Author(s):  
Maria Michaela Porzio
Keyword(s):  
Heat Equation ◽  
Parabolic Equations ◽  
Parabolic Problems ◽  
Summable Function ◽  
Time Behavior ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Evolution Problems ◽  
Degenerate Equations ◽  
Uniqueness Results

AbstractIn this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

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