scholarly journals Very Weak Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions

2009 ◽  
Vol 16 (2) ◽  
pp. 157-186
Author(s):  
Vuk Milišić
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Vertical Boundary

Convex integration solutions for the geometrically nonlinear two-well problem with higher Sobolev regularity

2020 ◽  
Vol 30 (03) ◽  
pp. 611-651
Author(s):  
Francesco Della Porta ◽  
Angkana Rüland
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Geometrically Nonlinear ◽  
Selection Mechanism ◽  
Matrix Space ◽  
Sobolev Regularity ◽  
Convex Integration ◽  
Nonlinear Matrix ◽  
Deformation Gradients ◽  
Higher Sobolev Regularity

In this paper, we discuss higher Sobolev regularity of convex integration solutions for the geometrically nonlinear two-well problem. More precisely, we construct solutions to the differential inclusion [Formula: see text] subject to suitable affine boundary conditions for [Formula: see text] with [Formula: see text] such that the associated deformation gradients [Formula: see text] enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape-memory alloys where [Formula: see text], and where the energy minimisers constructed by convex integration satisfy higher Sobolev regularity. We show that in spite of additional difficulties arising from the treatment of the nonlinear matrix space geometry, it is possible to deal with the geometrically nonlinear two-well problem within the framework outlined in [A. Rüland, C. Zillinger and B. Zwicknagl, Higher Sobolev regularity of convex integration solutions in elasticity: The Dirichlet problem with affine data in int[Formula: see text], SIAM J. Math. Anal. 50 (2018) 3791–3841]. Physically, our investigation of convex integration solutions at higher Sobolev regularity is motivated by viewing regularity as a possible selection mechanism of microstructures.

scholarly journals On the discreteness of the spectra of the Dirichlet and Neumannp-biharmonic problems

2004 ◽  
Vol 2004 (9) ◽  
pp. 777-792 ◽  
Author(s):  
Jiří Benedikt
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Neumann Problem ◽  
Nonlinear Boundary ◽  
Neumann Boundary ◽  
Unbounded Sequence ◽  
The Neumann Problem ◽  
Biharmonic Problems ◽  
Zero Points

We are interested in a nonlinear boundary value problem for(|u″|p−2u″)′​′=λ|u|p−2uin[0,1],p>1, with Dirichlet and Neumann boundary conditions. We prove that eigenvalues of the Dirichlet problem are positive, simple, and isolated, and form an increasing unbounded sequence. An eigenfunction, corresponding to thenth eigenvalue, has preciselyn−1zero points in(0,1). Eigenvalues of the Neumann problem are nonnegative and isolated,0is an eigenvalue which is not simple, and the positive eigenvalues are simple and they form an increasing unbounded sequence. An eigenfunction, corresponding to thenth positive eigenvalue, has preciselyn+1zero points in(0,1).

Nonlinear Dirichlet problem with non local regional diffusion

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
César E. Torres Ledesma
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Dirichlet Problem ◽  
Non Local ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractThe purpose of this paper is to study the existence of solutions for equations driven by a non-local regional operator with homogeneous Dirichlet boundary conditions. More precisely, we consider the problemwhere the nonlinear term

scholarly journals INTERVAL EXTENSION STRUCTURE BASIC TYPES OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

2018 ◽  
pp. 10-14
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Interval Function ◽  
The Third ◽  
Interval Extension ◽  
The Neumann Problem

. In this article, the interval expansion of the structure of solving basic types of boundary value problems for partial differential equations of the second order of making the basic operations that compose interval arithmetic is developed. For the differential equation (1) of the type, when constructing the interval expansion of the structure of the formula, structural formulas were used to construct with the Rfunction method and 4 problems were studied — the Dirichlet problem, the Neumann problem, the third type problem, the mixed boundary conditions problem. For the Dirichlet problem, the solution is an interval expansion of the structure in the form (5), where 𝑃 = {𝜔𝛷 , 𝜔𝛷̅, 𝜔̅𝛷, 𝜔̅𝛷̅} и [ 𝛷, 𝛷̅]is an indefinite interval function. For the Neumann problem, a solution is solved in the interval extension of the structure, [ 𝛷1, 𝛷1̅̅̅̅], [ 𝛷2, 𝛷2̅̅̅̅] is an indefinite interval function and 𝐷1 is a differential operator of the form. For the problem of the third type, the solution is solved in the interval extension of the structure, [ 𝛷1, 𝛷1̅̅̅̅], [𝛷2, 𝛷2̅̅̅̅] -indefinite, interval function, 𝐷1 - differential operator of the form (9). For the problem, mixed boundary conditions are treated. The solution In the interval extension of the structure,[ 𝛷1, 𝛷1], [ 𝛷2, 𝛷2̅̅̅̅] is an indefinite interval function and 𝐷1 is a differential operator of the form.

scholarly journals On fractional higher-order Dirichlet boundary value problems: Between the Laplacian and the bilaplacian

2021 ◽  
pp. 255-277
Author(s):  
Alberto Saldaña
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Natural Extension ◽  
Higher Order ◽  
Dirichlet Boundary Conditions ◽  
Maximum Principles ◽  
Limiting Behavior

The solutions of boundary value problems for the Laplacian and the bilaplacian exhibit very different qualitative behaviors. Particularly, the failure of general maximum principles for the bilaplacian implies that solutions of higher-order problems are less rigid and more complex. One way to better understand this transition is to study the intermediate Dirichlet problem in terms of fractional Laplacians. This survey aims to be an introduction to this type of problems; in particular, the different pointwise notions for these operators is introduced considering a suitable natural extension of the Dirichlet boundary conditions for the fractional setting. Solutions are obtained variationally and, in the case of the ball, via explicit kernels. The validity of maximum principles for these intermediate problems is also discussed as well as the limiting behavior of solutions when approaching the Laplacian or the bilaplacian case.

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