Enclosure theorems and barrier principles for energy stationary currents and the associated Brakke-flow

Analysis ◽  
2017 ◽  
Vol 37 (4) ◽  
Author(s):  
Patrick Henkemeyer
Keyword(s):  
Minimal Surfaces ◽  
Existence Theorems ◽  
Curvature Flow ◽  
Energy Functional ◽  
Weak Formulation ◽  
Geometric Properties ◽  
Comparison Principles ◽  
Gravitational Forces

AbstractWe discuss certain quantitative geometric properties of energy stationary currents describing minimal surfaces under gravitational forces. Enclosure theorems give statements about the confinement of the support of currents to certain enclosing sets on the basis that one knows something about the position of their boundaries. These results are closely related to non-existence theorems for currents with connected support. Finally, we define a weak formulation in the theory of varifolds for the curvature flow associated to this energy functional. We extend the enclosure results to the flow and discuss several comparison principles.

scholarly journals Nonsmooth critical point theory and nonlinear elliptic equations at resonance

2000 ◽  
Vol 69 (2) ◽  
pp. 245-271 ◽  
Author(s):  
Nikolaos C. Kourogenis ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Existence Theorems ◽  
Nonlinear Elliptic Equations ◽  
Point Theory ◽  
Energy Functional ◽  
Existence Theory ◽  
Nonsmooth Critical Point Theory ◽  
Nonsmooth Critical Point ◽  
The Right

AbstractIn this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.

scholarly journals Nonlinear Periodic Systems with thep-Laplacian: Existence and Multiplicity Results

2007 ◽  
Vol 2007 ◽  
pp. 1-23
Author(s):  
Francesca Papalini
Keyword(s):  
Existence Theorems ◽  
Point Theory ◽  
Energy Functional ◽  
Periodic Systems ◽  
Multiplicity Result ◽  
Nonsmooth Critical Point Theory ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Nonsmooth Critical Point ◽  
Locally Lipschitz

We study second-order nonlinear periodic systems driven by the vectorp-Laplacian with a nonsmooth, locally Lipschitz potential function. Under minimal and natural hypotheses on the potential and using variational methods based on the nonsmooth critical point theory, we prove existence theorems and a multiplicity result. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (unbounded from both above and below).

A p(x)-Laplacian extension of the Díaz-Saa inequality and some applications

2019 ◽  
Vol 150 (1) ◽  
pp. 205-232 ◽  
Author(s):  
Peter Takáč ◽  
Jacques Giacomoni
Keyword(s):  
Elliptic Equations ◽  
Energy Functional ◽  
Quasilinear Elliptic Equations ◽  
Uniqueness Of Solutions ◽  
Comparison Principles ◽  
Convexity Properties ◽  
Quasilinear Elliptic

AbstractThe main result of this work is a new extension of the well-known inequality by Díaz and Saa which, in our case, involves an anisotropic operator, such as the p(x)-Laplacian, $\Delta _{p(x)}u\equiv {\rm div}( \vert \nabla u \vert ^{p(x)-2}\nabla u)$. Our present extension of this inequality enables us to establish several new results on the uniqueness of solutions and comparison principles for some anisotropic quasilinear elliptic equations. Our proofs take advantage of certain convexity properties of the energy functional associated with the p(x)-Laplacian.

Geometric properties of minimal surfaces with free boundaries

1983 ◽  
Vol 184 (4) ◽  
pp. 497-509 ◽  
Author(s):  
Stefan Hildebrandt ◽  
Johannes C. C. Nitsche
Keyword(s):  
Minimal Surfaces ◽  
Free Boundaries ◽  
Geometric Properties

scholarly journals On a nonlocal Cahn–Hilliard model permitting sharp interfaces

2021 ◽  
pp. 1-38
Author(s):  
Olena Burkovska ◽  
Max Gunzburger
Keyword(s):  
Inequality Constraints ◽  
Coupled System ◽  
Energy Functional ◽  
Sharp Interface ◽  
Nonlocal Model ◽  
Weak Formulation ◽  
Diffuse Interfaces ◽  
Spatial Dimensions ◽  
Pure Phases ◽  
Sharp Interfaces

A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. Here, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.

scholarly journals Combined plasma–coil optimization algorithms

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
S. A. Henneberg ◽  
S. R. Hudson ◽  
D. Pfefferlé ◽  
P. Helander
Keyword(s):  
Free Boundary ◽  
Optimization Algorithms ◽  
Energy Functional ◽  
Equilibrium Calculation ◽  
Weak Formulation ◽  
Fixed Boundary ◽  
Minimization Principle ◽  
Boundary Equilibrium ◽  
Boundary Optimization ◽  
Coil Optimization

Combined plasma–coil optimization approaches for designing stellarators are discussed and a new method for calculating free-boundary equilibria for multiregion relaxed magnetohydrodynmics (MRxMHD) is proposed. Four distinct categories of stellarator optimization, two of which are novel approaches, are the fixed-boundary optimization, the generalized fixed-boundary optimization, the quasi-free-boundary optimization, and the free-boundary (coil) optimization. These are described using the MRxMHD energy functional, the Biot–Savart integral, the coil-penalty functional and the virtual casing integral and their derivatives. The proposed free-boundary equilibrium calculation differs from existing methods in how the boundary-value problem is posed, and for the new approach it seems that there is not an associated energy minimization principle because a non-symmetric functional arises. We propose to solve the weak formulation of this problem using a spectral-Galerkin method, and this will reduce the free-boundary equilibrium calculation to something comparable to a fixed-boundary calculation. In our discussion of combined plasma–coil optimization algorithms, we emphasize the importance of the stability matrix.

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