Bifurcation for non-differentiable operators with an application to elasticity

We consider operator equations of the form

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.     

A Locally Quadratic Glimm Functional and Sharp Convergence Rate of the Glimm Scheme for Nonlinear Hyperbolic Systems

Consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one space dimension

On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases

This paper is motivated by the study of a version of the so-called Schrödinger–Poisson–Slater problem: $- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,$ where ${u \in H^{1}(\mathbb {R}^3)}This paper is motivated by the study of a version of the so-called Schr?dinger–Poisson–Slater problem:

- Du + wu + l( u2 *\frac1|x| ) u=|u|p-2u,- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,      5. Asymptotic Stability of the Wave Equation on Compact Manifolds and Locally Distributed Damping: A Sharp Result      M. M. Cavalcanti  V. N. Domingos Cavalcanti  R. Fukuoka  J. A. Soriano 《Archive for Rational Mechanics and Analysis》2010,197(3):925-964 Let (M, g) be a n-dimensional ( ${n\geqq 2}Let (M, g) be a n-dimensional ( n\geqq 2{n\geqq 2}) compact Riemannian manifold with boundary where g denotes a Riemannian metric of class C ∞. This paper is concerned with the study of the wave equation on (M, g) with locally distributed damping, described by l utt - Dgu+ a(x) g(ut)=0,   on M×] 0,￥[ ,u=0 on ?M ×] 0,￥[, \left. \begin{array}{l} u_{tt} - \Delta_{{\bf g}}u+ a(x)\,g(u_{t})=0,\quad\hbox{on\ \thinspace}{M}\times \left] 0,\infty\right[ ,u=0\,\hbox{on}\,\partial M \times \left] 0,\infty \right[, \end{array} \right.      6. Nonpositive solutions of one functional differential inequality      A. Lomtatidze  Z. Opluštil  J. Šremr 《Nonlinear Oscillations》2009,12(4):474-509 We establish efficient conditions guaranteeing that every solution of the problem u￠(t) 3 l(u)(t),    u(a) 3 h(u), u'(t) \geq \ell (u)(t),\quad u(a) \geq h(u),      7. Positive Solutions for a Quasilinear Schr?dinger Equation with Critical Growth      Giovany M. Figueiredo  Marcelo F. Furtado 《Journal of Dynamics and Differential Equations》2012,24(1):13-28 We consider the quasilinear problem -ep\textdiv(|?u|p-2?u) + V(z)up-1 = f(u) + up*-1, u ? W1,p(\mathbbRN), -\varepsilon^p\text{div}(|\nabla u|^{p-2}\nabla u) + V(z)u^{p-1} = f(u) + u^{p^*-1},\,u \in W^{1,p}\left(\mathbb{R}^N\right),      8. Hydromagnetic stagnation point flow of a viscous fluid over?a?stretching or shrinking sheet      Robert A. Van Gorder  K. Vajravelu  I. Pop 《Meccanica》2012,47(1):31-50 We establish the existence and uniqueness results over the semi-infinite interval [0,∞) for a class of nonlinear third order ordinary differential equations of the form lf"￠( h) + f( h)f"( h) - ( f￠( h) )2 - Mf￠( h)    + C(C + M ) = 0,f( 0 ) = s ,       f￠( 0 ) = c,       limh? ￥ f￠( h) = C.\begin{array}{l}f'( \eta) + f( \eta)f'( \eta) - ( f'( \eta) )^{2} - Mf'( \eta)\\[6pt]\quad {}+ C(C + M ) = 0,\\[6pt]f( 0 ) = s ,\qquad f'( 0 ) = \chi ,\qquad \displaystyle\lim\limits_{\eta \to \infty} f'( \eta) = C.\end{array}      9. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales      S. H. Saker 《Nonlinear Oscillations》2011,13(3):407-428 Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional dynamic equation ( p(t)( [ y(t) + r(t)y( t(t) ) ]D )g )D + f( t,y( d(t) ) = 0,    t ? [ t0,￥ )\mathbbT, {\left( {p(t){{\left( {{{\left[ {y(t) + r(t)y\left( {\tau (t)} \right)} \right]}^\Delta }} \right)}^\gamma }} \right)^\Delta } + f\left( {t,y\left( {\delta (t)} \right)} \right. = 0,\quad t \in {\left[ {{t_0},\infty } \right)_\mathbb{T}},      10. Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation      Helge Holden  Xavier Raynaud 《Archive for Rational Mechanics and Analysis》2011,201(3):871-964 We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u tt − c(u)(c(u)u x ) x  = 0. We allow for initial data u| t = 0 and u t | t=0 that contain measures. We assume that 0 < k-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (ut2+c(u)2ux2)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.      11. Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks      Anna L. Mazzucato  Victor Nistor 《Archive for Rational Mechanics and Analysis》2010,195(1):25-73 We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces Km+1a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of Kma{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.      12. Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent      Zhijie Chen  Wenming Zou 《Archive for Rational Mechanics and Analysis》2012,205(2):515-551 In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l1 u = m1 u3+buv2,     x ? W,-Dv +l2 v = m2 v3+bvu2,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.      13. Boundary Regularity in Variational Problems      Jan Kristensen  Giuseppe Mingione 《Archive for Rational Mechanics and Analysis》2010,198(2):369-455 We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRn ? \mathbbRN{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      14. Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary      Jann-Long Chern  Chang-Shou Lin 《Archive for Rational Mechanics and Analysis》2010,197(2):401-432 Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in RN, N\geqq 3{{\bf R}^N, N\geqq 3}, and Da1,2(W){D_a^{1,2}(\Omega)} be the completion of C0￥(W){C_0^\infty(\Omega)} with respect to the norm: ||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      15. Isolated Singularity for the Stationary Navier—Stokes System      H. J. Choe 《Journal of Mathematical Fluid Mechanics》2000,2(2):151-184 In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in BR \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|-(n - 1)/2) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L2n/(n - 1)(BR) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? Lb(BR) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.      16. A Theory of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux      Boris Andreianov  Kenneth Hvistendahl Karlsen  Nils Henrik Risebro 《Archive for Rational Mechanics and Analysis》2011,201(1):27-86 We propose a general framework for the study of L 1 contractive semigroups of solutions to conservation laws with discontinuous flux: $ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \right.\quad\quad\quad (\rm CL) $ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \end{array} \right.\quad\quad\quad (\rm CL)      17. On a Volume‐Constrained Variational Problem      Luigi Ambrosio  Irene Fonseca  Paolo Marcellini  Luc Tartar 《Archive for Rational Mechanics and Analysis》1999,149(1):23-47 . Existence of minimizers for a volume-constrained energy $ E(u) := \int_{\Omega} W(\nabla u)\, dx Existence of minimizers for a volume-constrained energy E(u) : = òW W(?u) dx E(u) := \int_{\Omega} W(\nabla u)\, dx where LN({u = zi}) = ai, i = 1, ?, P, {\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P, is proved for the case in which ziz_i are extremal points of a compact, convex set in \Bbb Rd\Bbb R^d and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1d=1, P=2P=2, W(x)=|x|2W(\xi)=|\xi|^2, and the &-limit as the sum of the measures of the 2 phases tends to \L(W)\L(\Omega) is identified. Minimizers are fully characterized when N=1N=1, and candidates for solutions are studied for the circle and the square in the plane.      18. Existence Theory by Front Tracking for General Nonlinear Hyperbolic Systems      Fabio Ancona  Andrea Marson 《Archive for Rational Mechanics and Analysis》2007,185(2):287-340 We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension where , is a smooth matrix-valued map and the initial data is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate solutions converging in to the vanishing viscosity solution of (1), which, by the results in [6], is the unique limit of solutions to the (artificial) viscous parabolic approximationas . In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in , the limit of front tracking approximations provides a weak solution of the system of conservation laws u t + F(u) x = 0, satisfying the Liu admissibility conditions. These results are achieved under the only assumption of strict hyperbolicity of the matrices A(u), . In particular, our construction applies to general, strictly hyperbolic systems of conservation laws with characteristic fields that do not satisfy the standard conditions of genuine nonlinearity or of linear degeneracy in the sense of Lax[17], or in the generalized sense of Liu[23]. Dedicated to Prof. Tai Ping Liu on the occasion of his 60 th birthday      19. Positive Solutions for Semilinear Elliptic Equations: Two Simple Models with Several Bifurcations      Matteo Franca 《Journal of Dynamics and Differential Equations》2011,23(3):573-611 In this paper we analyze the structure of positive radial solutions for the following semi-linear equations: Du + f(u,|x|)=0 \Delta u + f(u,|{\bf x}|)=0      20. Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems      Jaume Llibre  Clàudia Valls 《Journal of Dynamics and Differential Equations》2010,22(4):657-675 We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-7-2j2 (A z5+j[`(z)]2+j + B z4+j[`(z)]3+j + C z3+j[`(z)]4+j+D[`(z)]7+2j ),\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Asymptotic Stability of the Wave Equation on Compact Manifolds and Locally Distributed Damping: A Sharp Result

Let (M, g) be a n-dimensional ( ${n\geqq 2}Let (M, g) be a n-dimensional ( n\geqq 2{n\geqq 2}) compact Riemannian manifold with boundary where g denotes a Riemannian metric of class C ^∞. This paper is concerned with the study of the wave equation on (M, g) with locally distributed damping, described by

l utt - Dgu+ a(x) g(ut)=0,   on M×] 0,￥[ ,u=0 on ?M ×] 0,￥[, \left. \begin{array}{l} u_{tt} - \Delta_{{\bf g}}u+ a(x)\,g(u_{t})=0,\quad\hbox{on\ \thinspace}{M}\times \left] 0,\infty\right[ ,u=0\,\hbox{on}\,\partial M \times \left] 0,\infty \right[, \end{array} \right.      6. Nonpositive solutions of one functional differential inequality      A. Lomtatidze  Z. Opluštil  J. Šremr 《Nonlinear Oscillations》2009,12(4):474-509 We establish efficient conditions guaranteeing that every solution of the problem u￠(t) 3 l(u)(t),    u(a) 3 h(u), u'(t) \geq \ell (u)(t),\quad u(a) \geq h(u),      7. Positive Solutions for a Quasilinear Schr?dinger Equation with Critical Growth      Giovany M. Figueiredo  Marcelo F. Furtado 《Journal of Dynamics and Differential Equations》2012,24(1):13-28 We consider the quasilinear problem -ep\textdiv(|?u|p-2?u) + V(z)up-1 = f(u) + up*-1, u ? W1,p(\mathbbRN), -\varepsilon^p\text{div}(|\nabla u|^{p-2}\nabla u) + V(z)u^{p-1} = f(u) + u^{p^*-1},\,u \in W^{1,p}\left(\mathbb{R}^N\right),      8. Hydromagnetic stagnation point flow of a viscous fluid over?a?stretching or shrinking sheet      Robert A. Van Gorder  K. Vajravelu  I. Pop 《Meccanica》2012,47(1):31-50 We establish the existence and uniqueness results over the semi-infinite interval [0,∞) for a class of nonlinear third order ordinary differential equations of the form lf"￠( h) + f( h)f"( h) - ( f￠( h) )2 - Mf￠( h)    + C(C + M ) = 0,f( 0 ) = s ,       f￠( 0 ) = c,       limh? ￥ f￠( h) = C.\begin{array}{l}f'( \eta) + f( \eta)f'( \eta) - ( f'( \eta) )^{2} - Mf'( \eta)\\[6pt]\quad {}+ C(C + M ) = 0,\\[6pt]f( 0 ) = s ,\qquad f'( 0 ) = \chi ,\qquad \displaystyle\lim\limits_{\eta \to \infty} f'( \eta) = C.\end{array}      9. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales      S. H. Saker 《Nonlinear Oscillations》2011,13(3):407-428 Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional dynamic equation ( p(t)( [ y(t) + r(t)y( t(t) ) ]D )g )D + f( t,y( d(t) ) = 0,    t ? [ t0,￥ )\mathbbT, {\left( {p(t){{\left( {{{\left[ {y(t) + r(t)y\left( {\tau (t)} \right)} \right]}^\Delta }} \right)}^\gamma }} \right)^\Delta } + f\left( {t,y\left( {\delta (t)} \right)} \right. = 0,\quad t \in {\left[ {{t_0},\infty } \right)_\mathbb{T}},      10. Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation      Helge Holden  Xavier Raynaud 《Archive for Rational Mechanics and Analysis》2011,201(3):871-964 We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u tt − c(u)(c(u)u x ) x  = 0. We allow for initial data u| t = 0 and u t | t=0 that contain measures. We assume that 0 < k-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (ut2+c(u)2ux2)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.      11. Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks      Anna L. Mazzucato  Victor Nistor 《Archive for Rational Mechanics and Analysis》2010,195(1):25-73 We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces Km+1a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of Kma{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.      12. Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent      Zhijie Chen  Wenming Zou 《Archive for Rational Mechanics and Analysis》2012,205(2):515-551 In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l1 u = m1 u3+buv2,     x ? W,-Dv +l2 v = m2 v3+bvu2,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.      13. Boundary Regularity in Variational Problems      Jan Kristensen  Giuseppe Mingione 《Archive for Rational Mechanics and Analysis》2010,198(2):369-455 We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRn ? \mathbbRN{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      14. Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary      Jann-Long Chern  Chang-Shou Lin 《Archive for Rational Mechanics and Analysis》2010,197(2):401-432 Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in RN, N\geqq 3{{\bf R}^N, N\geqq 3}, and Da1,2(W){D_a^{1,2}(\Omega)} be the completion of C0￥(W){C_0^\infty(\Omega)} with respect to the norm: ||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      15. Isolated Singularity for the Stationary Navier—Stokes System      H. J. Choe 《Journal of Mathematical Fluid Mechanics》2000,2(2):151-184 In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in BR \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|-(n - 1)/2) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L2n/(n - 1)(BR) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? Lb(BR) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.      16. A Theory of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux      Boris Andreianov  Kenneth Hvistendahl Karlsen  Nils Henrik Risebro 《Archive for Rational Mechanics and Analysis》2011,201(1):27-86 We propose a general framework for the study of L 1 contractive semigroups of solutions to conservation laws with discontinuous flux: $ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \right.\quad\quad\quad (\rm CL) $ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \end{array} \right.\quad\quad\quad (\rm CL)      17. On a Volume‐Constrained Variational Problem      Luigi Ambrosio  Irene Fonseca  Paolo Marcellini  Luc Tartar 《Archive for Rational Mechanics and Analysis》1999,149(1):23-47 . Existence of minimizers for a volume-constrained energy $ E(u) := \int_{\Omega} W(\nabla u)\, dx Existence of minimizers for a volume-constrained energy E(u) : = òW W(?u) dx E(u) := \int_{\Omega} W(\nabla u)\, dx where LN({u = zi}) = ai, i = 1, ?, P, {\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P, is proved for the case in which ziz_i are extremal points of a compact, convex set in \Bbb Rd\Bbb R^d and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1d=1, P=2P=2, W(x)=|x|2W(\xi)=|\xi|^2, and the &-limit as the sum of the measures of the 2 phases tends to \L(W)\L(\Omega) is identified. Minimizers are fully characterized when N=1N=1, and candidates for solutions are studied for the circle and the square in the plane.      18. Existence Theory by Front Tracking for General Nonlinear Hyperbolic Systems      Fabio Ancona  Andrea Marson 《Archive for Rational Mechanics and Analysis》2007,185(2):287-340 We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension where , is a smooth matrix-valued map and the initial data is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate solutions converging in to the vanishing viscosity solution of (1), which, by the results in [6], is the unique limit of solutions to the (artificial) viscous parabolic approximationas . In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in , the limit of front tracking approximations provides a weak solution of the system of conservation laws u t + F(u) x = 0, satisfying the Liu admissibility conditions. These results are achieved under the only assumption of strict hyperbolicity of the matrices A(u), . In particular, our construction applies to general, strictly hyperbolic systems of conservation laws with characteristic fields that do not satisfy the standard conditions of genuine nonlinearity or of linear degeneracy in the sense of Lax[17], or in the generalized sense of Liu[23]. Dedicated to Prof. Tai Ping Liu on the occasion of his 60 th birthday      19. Positive Solutions for Semilinear Elliptic Equations: Two Simple Models with Several Bifurcations      Matteo Franca 《Journal of Dynamics and Differential Equations》2011,23(3):573-611 In this paper we analyze the structure of positive radial solutions for the following semi-linear equations: Du + f(u,|x|)=0 \Delta u + f(u,|{\bf x}|)=0      20. Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems      Jaume Llibre  Clàudia Valls 《Journal of Dynamics and Differential Equations》2010,22(4):657-675 We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-7-2j2 (A z5+j[`(z)]2+j + B z4+j[`(z)]3+j + C z3+j[`(z)]4+j+D[`(z)]7+2j ),\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Nonpositive solutions of one functional differential inequality

We establish efficient conditions guaranteeing that every solution of the problem

Positive Solutions for a Quasilinear Schr?dinger Equation with Critical Growth

We consider the quasilinear problem

Hydromagnetic stagnation point flow of a viscous fluid over?a?stretching or shrinking sheet

We establish the existence and uniqueness results over the semi-infinite interval [0,∞) for a class of nonlinear third order ordinary differential equations of the form

Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales

Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional dynamic equation

Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u _tt − c(u)(c(u)u _x ) _x  = 0. We allow for initial data u| _{t = 0} and u _t | _t=0 that contain measures. We assume that 0 < k^-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (u_t²+c(u)²u_x²)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.     

Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks

We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces K^m+1_a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of K^m_a{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.     

Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent

In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l₁ u = m₁ u³+buv²,     x ? W,-Dv +l₂ v = m₂ v³+bvu²,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.     

Boundary Regularity in Variational Problems

We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRⁿ ? \mathbbR^N{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem

minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      14. Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary      Jann-Long Chern  Chang-Shou Lin 《Archive for Rational Mechanics and Analysis》2010,197(2):401-432 Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in RN, N\geqq 3{{\bf R}^N, N\geqq 3}, and Da1,2(W){D_a^{1,2}(\Omega)} be the completion of C0￥(W){C_0^\infty(\Omega)} with respect to the norm: ||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      15. Isolated Singularity for the Stationary Navier—Stokes System      H. J. Choe 《Journal of Mathematical Fluid Mechanics》2000,2(2):151-184 In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in BR \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|-(n - 1)/2) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L2n/(n - 1)(BR) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? Lb(BR) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.      16. A Theory of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux      Boris Andreianov  Kenneth Hvistendahl Karlsen  Nils Henrik Risebro 《Archive for Rational Mechanics and Analysis》2011,201(1):27-86 We propose a general framework for the study of L 1 contractive semigroups of solutions to conservation laws with discontinuous flux: $ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \right.\quad\quad\quad (\rm CL) $ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \end{array} \right.\quad\quad\quad (\rm CL)      17. On a Volume‐Constrained Variational Problem      Luigi Ambrosio  Irene Fonseca  Paolo Marcellini  Luc Tartar 《Archive for Rational Mechanics and Analysis》1999,149(1):23-47 . Existence of minimizers for a volume-constrained energy $ E(u) := \int_{\Omega} W(\nabla u)\, dx Existence of minimizers for a volume-constrained energy E(u) : = òW W(?u) dx E(u) := \int_{\Omega} W(\nabla u)\, dx where LN({u = zi}) = ai, i = 1, ?, P, {\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P, is proved for the case in which ziz_i are extremal points of a compact, convex set in \Bbb Rd\Bbb R^d and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1d=1, P=2P=2, W(x)=|x|2W(\xi)=|\xi|^2, and the &-limit as the sum of the measures of the 2 phases tends to \L(W)\L(\Omega) is identified. Minimizers are fully characterized when N=1N=1, and candidates for solutions are studied for the circle and the square in the plane.      18. Existence Theory by Front Tracking for General Nonlinear Hyperbolic Systems      Fabio Ancona  Andrea Marson 《Archive for Rational Mechanics and Analysis》2007,185(2):287-340 We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension where , is a smooth matrix-valued map and the initial data is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate solutions converging in to the vanishing viscosity solution of (1), which, by the results in [6], is the unique limit of solutions to the (artificial) viscous parabolic approximationas . In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in , the limit of front tracking approximations provides a weak solution of the system of conservation laws u t + F(u) x = 0, satisfying the Liu admissibility conditions. These results are achieved under the only assumption of strict hyperbolicity of the matrices A(u), . In particular, our construction applies to general, strictly hyperbolic systems of conservation laws with characteristic fields that do not satisfy the standard conditions of genuine nonlinearity or of linear degeneracy in the sense of Lax[17], or in the generalized sense of Liu[23]. Dedicated to Prof. Tai Ping Liu on the occasion of his 60 th birthday      19. Positive Solutions for Semilinear Elliptic Equations: Two Simple Models with Several Bifurcations      Matteo Franca 《Journal of Dynamics and Differential Equations》2011,23(3):573-611 In this paper we analyze the structure of positive radial solutions for the following semi-linear equations: Du + f(u,|x|)=0 \Delta u + f(u,|{\bf x}|)=0      20. Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems      Jaume Llibre  Clàudia Valls 《Journal of Dynamics and Differential Equations》2010,22(4):657-675 We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-7-2j2 (A z5+j[`(z)]2+j + B z4+j[`(z)]3+j + C z3+j[`(z)]4+j+D[`(z)]7+2j ),\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary

Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in R^N, N\geqq 3{{\bf R}^N, N\geqq 3}, and D_a^1,2(W){D_a^{1,2}(\Omega)} be the completion of C₀^￥(W){C_0^\infty(\Omega)} with respect to the norm:

||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      15. Isolated Singularity for the Stationary Navier—Stokes System      H. J. Choe 《Journal of Mathematical Fluid Mechanics》2000,2(2):151-184 In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in BR \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|-(n - 1)/2) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L2n/(n - 1)(BR) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? Lb(BR) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.      16. A Theory of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux      Boris Andreianov  Kenneth Hvistendahl Karlsen  Nils Henrik Risebro 《Archive for Rational Mechanics and Analysis》2011,201(1):27-86 We propose a general framework for the study of L 1 contractive semigroups of solutions to conservation laws with discontinuous flux: $ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \right.\quad\quad\quad (\rm CL) $ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \end{array} \right.\quad\quad\quad (\rm CL)      17. On a Volume‐Constrained Variational Problem      Luigi Ambrosio  Irene Fonseca  Paolo Marcellini  Luc Tartar 《Archive for Rational Mechanics and Analysis》1999,149(1):23-47 . Existence of minimizers for a volume-constrained energy $ E(u) := \int_{\Omega} W(\nabla u)\, dx Existence of minimizers for a volume-constrained energy E(u) : = òW W(?u) dx E(u) := \int_{\Omega} W(\nabla u)\, dx where LN({u = zi}) = ai, i = 1, ?, P, {\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P, is proved for the case in which ziz_i are extremal points of a compact, convex set in \Bbb Rd\Bbb R^d and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1d=1, P=2P=2, W(x)=|x|2W(\xi)=|\xi|^2, and the &-limit as the sum of the measures of the 2 phases tends to \L(W)\L(\Omega) is identified. Minimizers are fully characterized when N=1N=1, and candidates for solutions are studied for the circle and the square in the plane.      18. Existence Theory by Front Tracking for General Nonlinear Hyperbolic Systems      Fabio Ancona  Andrea Marson 《Archive for Rational Mechanics and Analysis》2007,185(2):287-340 We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension where , is a smooth matrix-valued map and the initial data is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate solutions converging in to the vanishing viscosity solution of (1), which, by the results in [6], is the unique limit of solutions to the (artificial) viscous parabolic approximationas . In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in , the limit of front tracking approximations provides a weak solution of the system of conservation laws u t + F(u) x = 0, satisfying the Liu admissibility conditions. These results are achieved under the only assumption of strict hyperbolicity of the matrices A(u), . In particular, our construction applies to general, strictly hyperbolic systems of conservation laws with characteristic fields that do not satisfy the standard conditions of genuine nonlinearity or of linear degeneracy in the sense of Lax[17], or in the generalized sense of Liu[23]. Dedicated to Prof. Tai Ping Liu on the occasion of his 60 th birthday      19. Positive Solutions for Semilinear Elliptic Equations: Two Simple Models with Several Bifurcations      Matteo Franca 《Journal of Dynamics and Differential Equations》2011,23(3):573-611 In this paper we analyze the structure of positive radial solutions for the following semi-linear equations: Du + f(u,|x|)=0 \Delta u + f(u,|{\bf x}|)=0      20. Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems      Jaume Llibre  Clàudia Valls 《Journal of Dynamics and Differential Equations》2010,22(4):657-675 We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-7-2j2 (A z5+j[`(z)]2+j + B z4+j[`(z)]3+j + C z3+j[`(z)]4+j+D[`(z)]7+2j ),\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Isolated Singularity for the Stationary Navier—Stokes System

In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in B_R \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|^{-(n - 1)/2}) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L^{2n/(n - 1)}(B_R) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? L^b(B_R) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.     

A Theory of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

We propose a general framework for the study of L ¹ contractive semigroups of solutions to conservation laws with discontinuous flux:

On a Volume‐Constrained Variational Problem

. Existence of minimizers for a volume-constrained energy $ E(u) := \int_{\Omega} W(\nabla u)\, dx Existence of minimizers for a volume-constrained energy E(u) : = ò_W W(?u) dx E(u) := \int_{\Omega} W(\nabla u)\, dx where L^N({u = z_i}) = a_i, i = 1, ?, P, {\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P, is proved for the case in which z_iz_i are extremal points of a compact, convex set in \Bbb R^d\Bbb R^d and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1d=1, P=2P=2, W(x)=|x|²W(\xi)=|\xi|^2, and the &-limit as the sum of the measures of the 2 phases tends to \L(W)\L(\Omega) is identified. Minimizers are fully characterized when N=1N=1, and candidates for solutions are studied for the circle and the square in the plane.     

Existence Theory by Front Tracking for General Nonlinear Hyperbolic Systems

We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension

Positive Solutions for Semilinear Elliptic Equations: Two Simple Models with Several Bifurcations

In this paper we analyze the structure of positive radial solutions for the following semi-linear equations:

Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems

We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R²{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as