scholarly journals Discontinuous boundary conditions and the Dirichlet problem

1923 ◽  
Vol 25 (3) ◽  
pp. 307-307 ◽  
Author(s):  
Norbert Wiener
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Discontinuous 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).

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