scholarly journals On a Fractional Version of a Murat Compactness Result and Applications

2021 ◽  
Vol 53 (3) ◽  
pp. 3158-3187
Author(s):  
Harbir Antil ◽  
Carlos N. Rautenberg ◽  
Armin Schikorra
Keyword(s):  
Compactness Result

scholarly journals Nodal solutions for the fractional Yamabe problem on Heisenberg groups

2019 ◽  
Vol 150 (2) ◽  
pp. 771-788 ◽  
Author(s):  
Alexandru Kristály
Keyword(s):  
Weak Solutions ◽  
Unitary Group ◽  
Original Problem ◽  
Laplacian Operator ◽  
Yamabe Problem ◽  
Heisenberg Groups ◽  
Nodal Solutions ◽  
Compactness Result ◽  
Homogeneous Dimension ◽  
Nodal Properties

AbstractWe prove that the fractional Yamabe equation ${\rm {\cal L}}_\gamma u = \vert u \vert ^{((4\gamma )/(Q-2\gamma ))}u$ on the Heisenberg group ℍn has [n + 1/2] sequences of nodal (sign-changing) weak solutions whose elements have mutually different nodal properties, where ${\rm {\cal L}}_\gamma $ denotes the CR fractional sub-Laplacian operator on ℍn, Q = 2n + 2 is the homogeneous dimension of ℍn, and $\gamma \in \bigcup\nolimits_{k = 1}^n [k,((kQ)/Q-1)))$. Our argument is variational, based on a Ding-type conformal pulling-back transformation of the original problem into a problem on the CR sphere S2n + 1 combined with a suitable Hebey-Vaugon-type compactness result and group-theoretical constructions for special subgroups of the unitary group U(n + 1).

scholarly journals Finite-difference approximation of free-discontinuity problems

2001 ◽  
Vol 131 (3) ◽  
pp. 567-595 ◽  
Author(s):  
Massimo Gobbino ◽  
Maria Giovanna Mora
Keyword(s):  
Finite Difference ◽  
Finite Differences ◽  
Pointwise Convergence ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Compactness Result ◽  
Free Discontinuity Problems ◽  
Non Local ◽  
Γ Convergence

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence and a compactness result, which implies, in particular, the convergence of minima and minimizers.

Martin's axiom in the model theory of LA

1980 ◽  
Vol 45 (1) ◽  
pp. 172-176
Author(s):  
W. Richard Stark
Keyword(s):  
Model Theory ◽  
Equivalence Theorem ◽  
Compactness Theorem ◽  
Compactness Result ◽  
First Order ◽  
Martin's Axiom ◽  
New Class ◽  
Martin’S Axiom ◽  
Order Language ◽  
Omitting Types

Working in ZFC + Martin's Axiom we develop a generalization of the Barwise Compactness Theorem which holds in languages of cardinality less than . Next, using this compactness theorem, an omitting types theorem for fewer than types is proved. Finally, in ZFC, we prove that this compactness result implies Martin's Axiom (the Equivalence Theorem). Our compactness theorem applies to a new class of theories—ccΣ-theories—which generalize the countable Σ-theories of Barwise's theorem. The Omitting Types Theorem and the Equivalence Theorem serve as examples illustrating the use of ccΣ-theories.Assume = (A, ε) or = (A, ε R1,…,Rm) where is admissible. L() is the first-order language with constants for elements of A and relation symbols for relations in . LA is A ⋂ L∞ω where the L of L∞ω is any language in A. A theory T in LA is consistent if there is no derivation in A of a contradiction from T. is LA with new constants ca for each a and A. The basic terms of consist of the constants of and the terms f(ca1,…,cam) built directly from constants using functions f of . The symbol t is used for basic terms. A theory T in LA is Σ if it is defined by a formula of L(). The formula φ⌝ is a logical equivalent of ¬φ defined by: (1) φ⌝ = ¬φ if φ is atomic; (2) (¬φ)⌝ = φ (3) (⋁φ∈Φ φ)⌝ = ⋀φ∈Φ φ⌝; (4) (⋀φ∈Φ φ) ⋁φ∈Φ φ⌝; (5) (∃χφ(x))⌝ ∀χφ⌝(x); ∀χφ(x))⌝ = ∃χφ⌝(x).

NEW RESULTS ON THE ASYMPTOTIC BEHAVIOR OF DIRICHLET PROBLEMS IN PERFORATED DOMAINSDIRICHLET PROBLEMS IN PERFORATED DOMAINS

1994 ◽  
Vol 04 (03) ◽  
pp. 373-407 ◽  
Author(s):  
GIANNI DAL MASO ◽  
ADRIANA GARRONI
Keyword(s):  
Adjoint Operator ◽  
Arbitrary Sequence ◽  
Symmetric Part ◽  
Dirichlet Problems ◽  
Limit Measure ◽  
Compactness Result ◽  
Perforated Domains ◽  
Measurable Coefficients ◽  
Open Set ◽  
Γ Convergence

Let A be a linear elliptic operator of the second order with bounded measurable coefficients on a bounded open set Ω of Rn and let (Ωh) be an arbitrary sequence of open subsets of Ω. We prove the following compactness result: there exist a subsequence, still denoted by (Ωh), and a positive Borel measure μ on Ω, not charging polar sets, such that, for every f∈H−1(Ω) the solutions [Formula: see text] of the equations Auh=f in Ωh, extended to 0 on Ω\Ωh, converge weakly in [Formula: see text] to the unique solution [Formula: see text] of the problem [Formula: see text] When A is symmetric, this compactness result is already known and was obtained by Γ-convergence techniques. Our new proof, based on the method of oscillating test functions, extends the result to the non-symmetric case. The new technique, which is completely independent of Γ-convergence, relies on the study of the behavior of the solutions [Formula: see text] of the equations [Formula: see text] where A* is the adjoint operator. We prove also that the limit measure μ does not change if A is replaced by A*. Moreover, we prove that µ depends only on the symmetric part of the operator A, if the coefficients of the skew-symmetric part are continuous, while an explicit example shows that μ may depend also on the skew-symmetric part of A, when the coefficients are discontinuous.

On compactness results for multi-scale convergence

1999 ◽  
Vol 129 (3) ◽  
pp. 467-476 ◽  
Author(s):  
Erik J. Balder
Keyword(s):  
Sobolev Space ◽  
Image Space ◽  
Orthogonal Complement ◽  
Young Measures ◽  
Type Result ◽  
Compactness Result ◽  
Relative Compactness ◽  
Multi Scale ◽  
Free Fields ◽  
Extension Result

Two relative compactness results for two-scale convergence in homogenization, due to G. Nguetseng, were recently extended to the multi-scale case by G. Allaire and M. Briane. Whereas their extension of Nguetseng's first result, which is in L2, is straightforward, their extension of his second result, which takes place in the Sobolev space H1, is quite complicated, even though it follows Nguetseng by using the fact that the image of H1 under the gradient mapping is the orthogonal complement of the set of divergence-free functions. Here a much simpler proof is provided by deriving the H1-type result from combining the first extension result with the fact that the above-mentioned image space is also the space of all rotation-free fields. Moreover, this approach reveals that the two results can be seen as corollaries of a fundamental relative compactness result for Young measures.

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