The behaviour of microstructures with small shears of the austenite-martensite interface in martensitic phase transformations

Let $\Omega \subset \Bbb{R}^2$ denote a bounded domain whose boundary $\partial \Omega$ is Lipschitz and contains a segment $\Gamma_0$ representing the austenite-twinned martensite interface. We prove $$\displaystyle{\inf_{{u\in \cal W}(\Omega)} \int_\Omega \varphi(\nabla u(x,y))dxdy=0}$$ for any elastic energy density $\varphi : \Bbb{R}^2 \rightarrow [0,\infty)$ such that $\varphi(0,\pm 1)=0$. Here ${\cal W}(\Omega)$ consists of all Lipschitz functions $u$ with $u=0$ on $\Gamma_0$ and $|u_y|=1$ a.e. Apart from the trivial case $\Gamma_0 \subset \reel \times \{a\},~a\in \Bbb{R}$, this result is obtained through the construction of suitable minimizing sequences which differ substantially for vertical and non-vertical segments.     

On the relaxation and the Lavrentieff phenomenon for variational integrals with pointwise measurable gradient constraints

Relaxation problems for a functional of the type are analyzed, where is a bounded smooth open subset of and g is a Carathéodory function. The admissible functions u are forced to satisfy a pointwise gradient constraint of the type for a.e. being, for every , a bounded convex subset of , in general varying with x not necessarily in a smooth way. The relaxed functionals and of G obtained letting u vary respectively in , the set of the piecewise C ¹-functions in , and in in the definition of G are considered. For both of them integral representation results are proved, with an explicit representation formula for the density of . Examples are proposed showing that in general the two densities are different, and that the one of is not obtained from g simply by convexification arguments. Eventually, the results are discussed in the framework of Lavrentieff phenomenon, showing by means of an example that deep differences occur between and . Results in more general settings are also obtained.Received: 18 December 2002, Accepted: 18 November 2003, Published online: 16 July 2004Mathematics Subject Classification (2000): 49J45, 49J10, 49J53This work is part of the European Research Training Network Homogenization and Multiple Scales (HMS 2000), under contract HPRN-2000-00109. It is also part of the 2003-G.N.A.M.P.A. Project Metodi Variazionali per Strutture Sottili, Frontiere Oscillanti ed Energie Vincolate.     

带梯度项的半线性椭圆方程整体解的存在性

By a sub-supersolution method and a perturbed argument,we show the existence of entire solutions for the semilinear elliptic problem-△u + a(x)|▽u|~q = λb(x)g(u),u 0,x ∈ R~N,lim(|x|→∞) u(x) = 0,where q ∈(1,2],λ 0,a and b are locally Holder continuous,a ≥0,b 0,(?)x∈ R~N,arid g ∈ C~1((0,∞),(0,∞)) which may be both possibly singular at zero and strongly unbounded at infinity.     

一类Kirchhoff型系统解的存在性

In this paper,we are interested in the existence of positive solutions for the Kirchhoff type problems{-(a_1 + b_1M_1(∫_?|▽u|~pdx))△_(_pu) = λf(u,v),in ?,-(a_2 + b_2M_2(∫?|▽v|~qdx))△_(_qv) = λg(u,v),in ?,u = v = 0,on ??,where 1 p,q N,M i:R_0~+→ R~+(i = 1,2) are continuous and increasing functions.λ is a parameter,f,g ∈ C~1((0,∞) ×(0,∞)) × C([0,∞) × [0,∞)) are monotone functions such that f_s,f_t,g_s,g_t ≥ 0,and f(0,0) 0,g(0,0) 0(semipositone).Our proof is based on the sub-and super-solutions techniques.     

Chemotaxis-fluild 系统的全局弱解

We investigate the existence of the global weak solution to the coupled Chemotaxisfluid system ■in a bounded smooth domain ??R~2. Here, r≥0 and μ 0 are given constants,?Φ∈L~∞(?) and g∈L~2((0, T); L_σ~2(?)) are prescribed functions. We obtain the local existence of the weak solution of the system by using the Schauder fixed point theorem. Furthermore, we study the regularity estimate of this system. Utilizing the regularity estimates, we obtain that the coupled Chemotaxis-fluid system with the initial-boundary value problem possesses a global weak solution.     

An Existence Result for a Class of Chemically Reacting Systems with Sign-Changing Weights

We prove the existence of positive solutions for the system$$\begin{align*}\begin{cases}-\Delta_{p} u =\lambda a(x){f(v)}{u^{-\alpha}},\qquad x\in \Omega,\\-\Delta_{q} v = \lambda b(x){g(u)}{v^{-\beta}},\qquad x\in \Omega,\\u = v =0, \qquad x\in\partial \Omega,\end{cases}\end{align*}$$where $\Delta_{r}z={\rm div}(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator and $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\alpha, \beta \in (0,1).$ Here $ a(x)$ and $ b(x)$ are $C^{1}$ sign-changingfunctions that maybe negative near the boundary and $f,g $ are $C^{1}$ nondecreasing functions, such that $f, g :\ [0,\infty)\to [0,\infty);$ $f(s)>0,$ $g(s)>0$ for $s> 0$, $\lim_{s\to\infty}g(s)=\infty$ and$$\lim_{s\to\infty}\frac{f(Mg(s)^{\frac{1}{q-1}})}{s^{p-1+\alpha}}=0,\qquad \forall M>0.$$We discuss the existence of positive weak solutions when $f$, $g$, $a(x)$ and $b(x)$ satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.     

具有凹凸非线性项和变号位势函数拟线性椭圆系统解的多重结果

研究拟线性椭圆系统(?)的非平凡非负解或正解的多重性,这里Ω(?)R~N是具有光滑边界(?)Ω的有界域,1≤qp~*/p~*-q,其中当N≤p时,p~*=+∞,而当1相似文献   

A global compactness result for singular elliptic problems involving critical Sobolev exponent

Let be a bounded domain such that . Let be a (P.S.) sequence of the functional . We study the limit behaviour of and obtain a global compactness result.

     

Existence of solutions for a degenerate auasilinear elliptic system in bounded domain

Using variational methods, we study the existence of weak solutions forthe degenerate quasilinear elliptic system$$\left\{\begin{array}{ll}- \mathrm{div}\Big(h_1(x)|\nabla u|^{p-2}\nabla u\Big) = F_{u}(x,u,v) &\text{ in } \Omega,\\-\mathrm{div}\Big(h_2(x)|\nabla v|^{q-2}\nabla v\Big) = F_{v}(x,u,v) &\text{ in } \Omega,\\u=v=0 & \textrm{ on } \partial\Omega,\end{array}\right.$$where $\Omega\subset \mathbb R^N$ is a smooth bounded domain, $\nabla F= (F_u,F_v)$ stands for the gradient of $C^1$-function $F:\Omega\times\mathbb R^2 \to \mathbb R$, the weights $h_i$, $i=1,2$ are allowed to vanish somewhere,the primitive $F(x,u,v)$ is intimately related to the first eigenvalue of acorresponding quasilinear system.     

Entropy solutions to nonlinear elliptic anisotropic problem with variable exponent

In this work, we give an existence result of entropy solutions for nonlinear anisotropic elliptic equation of the type $$- \mbox{div} \big( a(x,u,\nabla u)\big)+ g(x,u,\nabla u) + |u|^{p_{0}(x)-2}u = f-\mbox{div} \phi(u),\quad \mbox{ in } \Omega,$$ where $-\mbox{div}\big(a(x,u,\nabla u)\big)$ is a Leray-Lions operator, $\phi \in C^{0}(I\!\!R,I\!\!R^{N})$. The function $g(x,u,\nabla u)$ is a nonlinear lower order term with natural growth with respect to $|\nabla u|$, satisfying the sign condition and the datum $f$ belongs to $L^1(\Omega)$.     

Sequential weak continuity of null Lagrangians at the boundary

We show weak* in measures on $\bar{\Omega }$ / weak- $L^1$ sequential continuity of $u\mapsto f(x,\nabla u):W^{1,p}(\Omega ;\mathbb{R }^m)\rightarrow L^1(\Omega )$ , where $f(x,\cdot )$ is a null Lagrangian for $x\in \Omega $ , it is a null Lagrangian at the boundary for $x\in \partial \Omega $ and $|f(x,A)|\le C(1+|A|^p)$ . We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why $u\mapsto \det \nabla u:W^{1,n}(\Omega ;\mathbb{R }^n)\rightarrow L^1(\Omega )$ fails to be weakly continuous. Further, we state a new weak lower semicontinuity theorem for integrands depending on null Lagrangians at the boundary. The paper closes with an example indicating that a well-known result on higher integrability of determinant by Müller (Bull. Am. Math. Soc. New Ser. 21(2): 245–248, 1989) need not necessarily extend to our setting. The notion of quasiconvexity at the boundary due to J.M. Ball and J. Marsden is central to our analysis.     

A note on Meyers' Theorem in

Lower semicontinuity properties of multiple integrals

are studied when may grow linearly with respect to the highest-order derivative, and admissible sequences converge strongly in It is shown that under certain continuity assumptions on convexity, -quasiconvexity or -polyconvexity of

ensures lower semicontinuity. The case where is -quasiconvex remains open except in some very particular cases, such as when

     

Extremal Functions for Adams Inequalities in Dimension Four

Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer $\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$, where $E_{\lambda_i(\Omega)}$ is eigenfunction space associated with $\lambda_i(\Omega)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(\Omega)$. For $0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we define a norm by $\|u\|_{2,\alpha}^{2}=\|\Delta u\|^2_2-\alpha \|u\|^2_2$ for $u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities$$\sup_{u\in E_{\ell}^{\bot},\,\| u\|_{2,\alpha}\leq 1}\int_{\Omega}e^{32\pi^2u^2}{\rm d}x<+\infty;$$moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overline{\Omega})$ with $\|u_0\|_{2,\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).     

A Baire's category method for the Dirichlet problem of quasiregular mappings

We adopt the idea of Baire's category method as presented in a series of papers by Dacorogna and Marcellini to study the boundary value problem for quasiregular mappings in space. Our main result is to prove that for any 0$"> and any piece-wise affine map with for almost every there exists a map such that

The theorems of Dacorogna and Marcellini do not directly apply to our result since the involved sets are unbounded. Our proof is elementary and does not require any notion of polyconvexity, quasiconvexity or rank-one convexity in the vectorial calculus of variations, as required in the papers by the quoted authors.

     

含有Sobolev-Hardy临界指标的奇异椭圆方程Neumann问题无穷多解的存在性

该文研究了如下的奇异椭圆方程Neumann问题$\left\{\begin{array}{ll}\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,\ \ &;x\in\Omega,\\D_\gamma{u}+\alpha(x)u=0,&;x\in\partial\Omega\backslash\{0\},\end{array}\right.$其中$\Omega $ 是 $ R^N$ 中具有 $ C^1$边界的有界区域, $ 0\in\partial\Omega$, $N\ge5$. $2^{*}(s)=\frac{2(N-s)}{N-2}$ (该文研究了如下的奇异椭圆方程Neumann问题$\left\{\begin{array}{ll}\disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u,\ \ &;x\in\Omega,\\D_\gamma{u}+\alpha(x)u=0,&;x\in\partial\Omega\backslash\{0\},\end{array}\right.$其中$\Omega $ 是 $ R^N$ 中具有 $ C^1$边界的有界区域, $ 0\in\partial\Omega$, $N\ge5$. $2^{*}(s)=\frac{2(N-s)}{N-2}$ (该文研究了如下的奇异椭圆方程Neumann问题其中Ω是RN中具有C1边界的有界区域,0∈■Ω,N≥5.2*(s)=2(N-s)/N-2(0≤s≤2)是临界Sobolev-Hardy指标, 10.利用变分方法和对偶喷泉定理,证明了这个方程无穷多解的存在性.     

Infinitely many solutions for a nonlocal problem

Consider a class of nonlocal problems $$ \left \{\begin{array}{ll} -(a-b\int_{\Omega}|\nabla u|^2dx)\Delta u= f(x,u),& \textrm{$x \in\Omega$},\u=0, & \textrm{$x \in\partial\Omega$}, \end{array} \right. $$ where $a>0, b>0$,~$\Omega\subset \mathbb{R}^N$ is a bounded open domain, $f:\overline{\Omega} \times \mathbb R \longrightarrow \mathbb R $ is a Carath$\acute{\mbox{e}}$odory function. Under suitable conditions, the equivariant link theorem without the $(P.S.)$ condition due to Willem is applied to prove that the above problem has infinitely many solutions, whose energy increasingly tends to $a^2/(4b)$, and they are neither large nor small.     

Superposition operator in Sobolev spaces on domains

For an arbitrary open set we characterize all functions on the real line such that for all . New element in the proof is based on Maz'ya's capacitary criterion for the imbedding .     

On a class of sublinear quasilinear elliptic problems

We establish existence and multiplicity of positive solutions to the quasilinear boundary value problem

where is a bounded domain in with smooth boundary , is continuous and p-sublinear at and is a large parameter.

     

A Singular Parabolic Boundary Value Problem in Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

We consider the operator ${\cal A}$ formally defined by ${\cal A}u(x)=\alpha(x)\Delta u(x)$ for any $x$ in a sufficiently smooth bounded open set $\Om\subset\R^N$ ($N\ge 1$), where $\alpha\in C(\ov\Omega)$ is a continuous nonnegative function vanishing only on $\partial\Omega$, and such that $1/\alpha$ is integrable in $\Omega$. We prove that the realization $A_p$ of ${\cal A}$, equipped with suitable nonlinear boundary conditions is an m-dissipative operator in suitably weighted $L^p(\Omega)$-spaces in the case where either $(p,N)\in (1,+\infty)\times\{1\}$ or $(p,N)=\{2\}\times\N$. Moreover, we prove that $A_p$ is a densely defined closed operator. We consider nonlinear boundary conditions of the following type: in the one dimensional case we take $\Omega=(0,1)$ and we assume that $u(j)=(-1)^j\beta_j(u(j))$ ($j=0,1$), $\beta_0$ and $\beta_1$ being nondecreasing continuous functions in $\R$ such that $\beta_0(0)=\beta_1(0)=0$; in the $N$-dimensional setting we assume that $(D_{\nu}u)_{|\partial\Omega}=-\beta(u_{|\partial\Omega})$, $\beta$ being a nondecreasing Lipschitz continuous function in $\R$ such that $\beta(0)=0$. Here $\nu$ denotes the unit outward normal to $\partial\Om$.     

On the <Emphasis Type="Italic">p</Emphasis>-Laplacian with Robin boundary conditions and boundary trace theorems

Let \(\Omega \subset \mathbb {R}^\nu \), \(\nu \ge 2\), be a \(C^{1,1}\) domain whose boundary \(\partial \Omega \) is either compact or behaves suitably at infinity. For \(p\in (1,\infty )\) and \(\alpha >0\), define

$$\begin{aligned} \Lambda (\Omega ,p,\alpha ):=\inf _{\begin{array}{c} u\in W^{1,p}(\Omega )\\ u\not \equiv 0 \end{array}}\dfrac{\displaystyle \int _\Omega |\nabla u|^p \mathrm {d} x - \alpha \displaystyle \int _{\partial \Omega } |u|^p\mathrm {d}\sigma }{\displaystyle \int _\Omega |u|^p\mathrm {d} x}, \end{aligned}$$

where \(\mathrm {d}\sigma \) is the surface measure on \(\partial \Omega \). We show the asymptotics

$$\begin{aligned} \Lambda (\Omega ,p,\alpha )=-(p-1)\alpha ^{\frac{p}{p-1}} - (\nu -1)H_\mathrm {max}\, \alpha + o(\alpha ), \quad \alpha \rightarrow +\infty , \end{aligned}$$

where \(H_\mathrm {max}\) is the maximum mean curvature of \(\partial \Omega \). The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities.