Lower semicontinuity of variational integrals on elliptic complexes

Anna Verde

Displaying similar documents to “Lower semicontinuity of variational integrals on elliptic complexes”

Wiener criterion for degenerate elliptic obstacle problem

Marco Biroli, Umberto Mosco (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

We give a Wiener criterion for the continuity of an obstacle problem relative to an elliptic degenerate problem with a weight in the $A_{2}$ class.

The largest prime factor of X³ + 2

A. J. Irving (2015)

Acta Arithmetica

Similarity:

Improving on a theorem of Heath-Brown, we show that if X is sufficiently large then a positive proportion of the values n³ + 2: n ∈ (X,2X] have a prime factor larger than $X^{1 + 10^{- 52}}$ .

Boundedness results of solutions to the equation $x^{\prime\prime\prime}+ax^{\prime\prime}+g(x)x^{\prime}+h(x)=p(t)$ without the hypothesis $h(x)\,\operatorname{sgn}x\geq 0$ for $|x|>R$ .

Ján Andres (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Per l'equazione differenziale ordinaria non lineare del 3° ordine indicata nel titolo, studiata da numerosi autori sotto l'ipotesi $h(x)\,\text{sgn}x\geq 0$ $for|x|>R$ , si dimostra l'esistenza di almeno una soluzione limitata sopprimendo l'ipotesi suddetta.

On the behaviour of solutions of the differential equations ${(r (t) y^{''})}^{'} + q (t) {(y^{'})}^{β} + p (t) y^{α} = f (t)$

N. Parhi, S. Parhi (1986)

Annales Polonici Mathematici

Similarity:

Optimal $L^{p}$ -properties of Green’s functions for non-divergence elliptic equations in two dimensions

Gioconda Moscariello, Carlo Sbordone (2005)

Studia Mathematica

Similarity:

A sharp integrability result for non-negative adjoint solutions to planar non-divergence elliptic equations is proved. A uniform estimate is also given for the Green's function.

Shifted values of the largest prime factor function and its average value in short intervals

Jean-Marie De Koninck, Imre Kátai (2016)

Colloquium Mathematicae

Similarity:

We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_{q} (n)$ stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence ${(α s_{q} (P (n)))}_{n \in ℕ}$ is uniformly distributed modulo 1.

Periodic solutions of $x^{''} + f (x) x^{' 2 n} + g (x) = μ p (t)$

S. Sędziwy (1969)

Annales Polonici Mathematici

Similarity:

Boundedness results of solutions to the equation $x^{\prime\prime\prime}+ax^{\prime\prime}+g(x)x^{\prime}+h(x)=p(t)$ without the hypothesis $h(x)\,\text{sgn}x\geq 0$ $for|x|>R$ .

Ján Andres (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

Periodic solutions to a non-linear parametric differential equation of the third order

Jan Andres, Jan Vorácek (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Si dimostra un teorema di esistenza di soluzioni periodiche dell'equazione differenziale ordinaria del terzo ordine $x^{\prime\prime\prime}+a(t,x,x^{\prime},x^{\prime\prime})x^{\prime\prime}+b(t,% x,x^{\prime},x^{\prime\prime})x^{\prime}+h(x)=e(t,x,x^{\prime},x^{\prime\prime})$ con le funzioni $a$ , $b$ , $e$ periodiche in $t$ di periodo $\omega$ .

Wiener criterion for degenerate elliptic obstacle problem

Marco Biroli, Umberto Mosco (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

We give a Wiener criterion for the continuity of an obstacle problem relative to an elliptic degenerate problem with a weight in the $A_{2}$ class.

$L^{p, Φ}$ spaces and their application to elliptic partial differential operators

Magdalena Jaroszewska (1983)

Annales Polonici Mathematici

Similarity:

On the non-convergence of successive approximations in the Darboux problem for the equation $z_{x y}^{''} = f (x, y, z)$

M. Kisielewicz (1977)

Colloquium Mathematicae

Similarity:

Periodic solutions to a non-linear parametric differential equation of the third order

Jan Andres, Jan Vorácek (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

T-p(x)-solutions for nonlinear elliptic equations with an L¹-dual datum

El Houssine Azroul, Abdelkrim Barbara, Meryem El Lekhlifi, Mohamed Rhoudaf (2012)

Applicationes Mathematicae

Similarity:

We establish the existence of a T-p(x)-solution for the p(x)-elliptic problem $- d i v (a (x, u, \nabla u)) + g (x, u) = f - d i v F$ in Ω, where Ω is a bounded open domain of $ℝ^{N}$ , N ≥ 2 and $a : Ω \times ℝ \times ℝ^{N} \to ℝ^{N}$ is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side f lies in L¹(Ω) and F belongs to $\prod_{i = 1}^{N} L^{p^{'} (\cdot)} (Ω)$ .

On Neumann boundary value problems for elliptic equations

Dimitrios A. Kandilakis (2004)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarity:

We provide two existence results for the nonlinear Neumann problem ⎧-div(a(x)∇u(x)) = f(x,u) in Ω ⎨ ⎩∂u/∂n = 0 on ∂Ω, where Ω is a smooth bounded domain in $ℝ^{N}$ , a is a weight function and f a nonlinear perturbation. Our approach is variational in character.

On the average value of the canonical height in higher dimensional families of elliptic curves

Wei Pin Wong (2014)

Acta Arithmetica

Similarity:

Given an elliptic curve E over a function field K = ℚ(T₁,...,Tₙ), we study the behavior of the canonical height $h ̂_{E_{ω}}$ of the specialized elliptic curve $E_{ω}$ with respect to the height of ω ∈ ℚⁿ. We prove that there exists a uniform nonzero lower bound for the average of the quotient $(h ̂_{E_{ω}} (P_{ω})) / h (ω)$ over all nontorsion P ∈ E(K).

Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

Zonghu Xiu, Caisheng Chen (2013)

Annales Polonici Mathematici

Similarity:

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ $- {d i v (| x |}^{- a p} {| \nabla u |}^{p - 2} {\nabla u) + h (x) | u |}^{p - 2} u = g (x) {| u |}^{r - 2} u$ , x ∈ Ω, ⎨ ⎩ ${| x |}^{- a p} {| \nabla u |}^{p - 2} \partial u / \partial ν = λ f (x) {| u |}^{q - 2} u$ , x ∈ ∂Ω, where Ω is an exterior domain in $ℝ^{N}$ , that is, $Ω = ℝ^{N} ∖ D$ , where D is a bounded domain in $ℝ^{N}$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function...

Convex integration and the $L^{p}$ theory of elliptic equations

Kari Astala, Daniel Faraco, László Székelyhidi Jr. (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

This paper deals with the $L^{p}$ theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general $L^{p}$ theory, developed in [1, 24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions...

Lower and upper bounds for the number of solutions of $p + h = P_{r}$

Kan Jiahai (1990)

Acta Arithmetica

Similarity:

$E_{1}$ -degeneration and $d^{'} d^{''}$ -lemma

Tai-Wei Chen, Chung-I Ho, Jyh-Haur Teh (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

For a double complex $(A, d^{'}, d^{''})$ , we show that if it satisfies the $d^{'} d^{''}$ -lemma and the spectral sequence ${E_{r}^{p, q}}$ induced by $A$ does not degenerate at $E_{0}$ , then it degenerates at $E_{1}$ . We apply this result to prove the degeneration at $E_{1}$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d^{'} d^{''}$ -lemma.

On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

Volodymyr Mykhaylyuk, Anatolij Plichko (2015)

Colloquium Mathematicae

Similarity:

We comment on a problem of Mazur from “The Scottish Book" concerning second partial derivatives. We prove that if a function f(x,y) of real variables defined on a rectangle has continuous derivative with respect to y and for almost all y the function $F_{y} (x) : = f_{y}^{'} (x, y)$ has finite variation, then almost everywhere on the rectangle the partial derivative $f_{y x}^{''}$ exists. We construct a separately twice differentiable function whose partial derivative $f_{x}^{'}$ is discontinuous with respect to the second variable on a...

A compactness result in thin-film micromagnetics and the optimality of the Néel wall

Radu Ignat, Felix Otto (2008)

Journal of the European Mathematical Society

Similarity:

In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^{1}$ -valued maps $m^{'}$ (the magnetization) of two variables $x^{'}$ : $E_{ε} (m^{'}) = ε \int {| \nabla^{'} \cdot m^{'} |}^{2} d x^{'} + \frac{1}{2} \int {|| \nabla^{'} |^{- 1 / 2} \nabla^{'} \cdot m^{'}|}^{2} d x^{'}$ . We are interested in the behavior of minimizers as $ε \to 0$ . They are expected to be $S^{1}$ -valued maps $m^{'}$ of vanishing distributional divergence $\nabla^{'} \cdot m^{'} = 0$ , so that appropriate boundary conditions enforce line discontinuities. For finite $ε > 0$ , these line discontinuities are approximated by smooth transition layers, the so-called Néel...

On hyperbolic partial differential equations in Banach spaces

Bogdan Rzepecki (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Viene dimostrata l'esistenza di soluzioni del problema di Darboux per l'equazione iperbolica $z^{\prime\prime}_{xy}=f(x,y,z,Z^{\prime}_{x},z_{y})$ sul planiquarto $x\geq 0$ , $y\geq 0$ . Qui, $f$ è una funzione continua, con valori in uno spazio Banach che soddisfano alcune condizioni di regolarità espresse in termini della misura di non-compattezza $\alpha$ .