Search results for "Maximum Principle"
showing 10 items of 56 documents
FINITE ELEMENT APPROXIMATION OF NONLOCAL HEAT RADIATION PROBLEMS
1998
This paper focuses on finite element error analysis for problems involving both conductive and radiative heat transfers. The radiative heat exchange is modeled with a nonlinear and nonlocal term that also makes the problem non-monotone. The continuous problem has a maximum principle which suggests the use of inverse monotone discretizations. We also estimate the error due to the approximation of the boundary by showing continuous dependence on the geometric data for the continuous problem. The final result of this paper is a rigorous justification and error analysis for methods that use the so-called view factors for numerical modeling of the heat radiation.
Acute Type Refinements of Tetrahedral Partitions of Polyhedral Domains
2001
We present a new technique to perform refinements on acute type tetrahedral partitions of a polyhedral domain, provided that the center of the circumscribed sphere around each tetrahedron belongs to the tetrahedron. The resulting family of partitions is of acute type; thus, all the tetrahedra satisfy the maximum angle condition. Both these properties are highly desirable in finite element analysis.
Comparison of Numerical Methods in the Contrast Imaging Problem in NMR
2013
International audience; In this article, the contrast imaging problem in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. A first synthesis of locally optimal solutions is given in the single-input case using geometric methods based on Pontryagin's maximum principle. We then compare these results using direct methods and a moment-based approach, and make a first step towards global optimality. Finally, some preliminary results are given in the bi-input case.
Maximum Principle and Application to Nuclear Magnetic Resonance and Magnetic Resonance Imaging
2018
In this section we state the Pontryagin maximum principle and we outline the proof. We adopt the presentation from Lee and Markus [64] where the result is presented into two theorems.
Regularity properties for quasiminimizers of a $(p,q)$-Dirichlet integral
2021
Using a variational approach we study interior regularity for quasiminimizers of a $(p,q)$-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincar\'{e} inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally H\"{o}lder continuous and they satisfy Harnack inequality, the strong maximum principle, and Liouville's Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for H\"older continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we cons…
Optimal Control of the Lotka-Volterra Equations with Applications
2022
In this article, the Lotka-Volterra model is analyzed to reduce the infection of a complex microbiote. The problem is set as an optimal control problem, where controls are associated to antibiotic or probiotic agents, or transplantations and bactericides. Candidates as minimizers are selected using the Maximum Principle and the closed loop optimal solution is discussed. In particular a 2d-model is constructed with 4 parameters to compute the optimal synthesis using homotopies on the parameters.
A singular (p,q)-equation with convection and a locally defined perturbation
2021
We consider a parametric Dirichlet problem driven by the (p,q)-Laplacian and a reaction which is gradient dependent (convection) and the competing effects of two more terms, one a parametric singular term and a locally defined perturbation. We show that for all small values of the parameter the problem has a positive smooth solution.
Principal eigenvalue of a very badly degenerate operator and applications
2007
Abstract In this paper, we define and investigate the properties of the principal eigenvalue of the singular infinity Laplace operator Δ ∞ u = ( D 2 u D u | D u | ) ⋅ D u | D u | . This operator arises from the optimal Lipschitz extension problem and it plays the same fundamental role in the calculus of variations of L ∞ functionals as the usual Laplacian does in the calculus of variations of L 2 functionals. Our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of Berestycki, Nirenberg and Varadhan [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operator…
Generalized Harnack inequality for semilinear elliptic equations
2015
Abstract This paper is concerned with semilinear equations in divergence form div ( A ( x ) D u ) = f ( u ) , where f : R → [ 0 , ∞ ) is nondecreasing. We introduce a sharp Harnack type inequality for nonnegative solutions which is a quantified version of the condition for strong maximum principle found by Vazquez and Pucci–Serrin in [30] , [24] and is closely related to the classical Keller–Osserman condition [15] , [22] for the existence of entire solutions.
Positive solutions for the Neumann p-Laplacian
2017
We examine parametric nonlinear Neumann problems driven by the p-Laplacian with asymptotically ( $$p-1$$ )-linear reaction term f(z, x) (as $$x\rightarrow +\infty $$ ). We determine the existence, nonexistence and minimality of positive solutions as the parameter $$\lambda >0$$ varies.