Determination of the solutions of boundary value problems for stationary stokes and Navier-Stokes equations having an unbounded Dirichlet integral

In unbounded domains Ω of the three-dimensional Euclidean space, having several outlets Ω _i i=1,...,N, at infinity, one investigates the time-dependent solutions of the Stokes and Navier-Stokes system of equations for incompressible fluids, equal to zero at the boundary of the domain Ω and having arbitrary flows ∝ _i through each of the outlets Ω_i (the numbers ∝ _i satisfy only the necessary condition: \(\sum\limits_{i = 1}^N {\alpha _i } = 0\) ). For these solutions one establishes Phragmén-Lindelöf and Saint-Venant type theorems characterizing the growth of solutions at infinity. On their basis, one formulates well-posed boundary value problems for the above indicated systems and domain Ω and one proves their solvability for any quantities ∝ _i . One investigates various properties of these solutions and one gives sufficient conditions for uniqueness theorems. In particular, when Ω is a pipe with cylindrical ends, then our solutions approach the Poiseuille flows with a given flow ∝ _i , for any in the case of the Stokes system and for ∝ _i smaller in absolute value than some critical value ∝ _i ^* , in the case of the Navier-Stokes system.     

Asymptotical self-similarity in nonlinear parabolic systems

We prove that blowing up solutions of the system

Existence and symmetry results for competing variational systems

In this paper we consider a class of gradient systems of type $$\begin{array}{ll} -c_{i} \Delta u_{i} + V_{i}(x)u_{i} = P_{u_i}(u),\qquad u_{1}, \ldots, u_{k} >\; 0\; \text{in}\; \Omega,\\ \quad u_{1} = \cdots = u_{k} = 0 \text{ on } \partial \Omega, \end{array}$$ in a bounded domain ${\Omega \subseteq \mathbb{R}^N}$ . Under suitable assumptions on V _i and P, we prove the existence of ground-state solutions for this problem. Moreover, for k = 2, assuming that the domain Ω and the potentials V _i are radially symmetric, we prove that the ground state solutions are foliated Schwarz symmetric with respect to antipodal points. We provide several examples for our abstract framework.     

Genericity of hyperbolicity and saddle-node bifurcations in reaction-diffusion equations depending on a parameter

Semilinear elliptic equations of the form $$\begin{array}{*{20}c} {\sum\limits_{i,j = 1}^n {(a_{ij} (x)u_{xi} (x))_{x_j } + } f(\lambda ,x,u(x)) = 0,} & {x \in \Omega ,} \\ {u(x) = 0,} & x \\ \end{array} $$ are considered, where λ ε ? is a parameter, Ω ? ? ⁿ is a bounded domain andf is a smooth non-linear function. It is shown that for ‘generic’ functionsf, the set of non-trivial solutions (λ,u) consists of a finite, or countable, collection of smooth, 1-dimensional curves and any such solution is either hyperbolic or is a saddle-node bifurcation point of the curve.     

Discontinuous superlinear elliptic equations of divergence form

We prove existence and global Hölder regularity of the weak solution to the Dirichlet problem $\left\{ {\begin{array}{lc} {{\rm div} \left( a^{ij} (x,u)D_{j} u \right) = b(x,u,Du) \quad {\rm in}\, \Omega \subset {\mathbb R}^{n}, \, n \ge 2,} \\ {u = 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,{\rm on}\, \partial\Omega \in C^{1}. } \\ \end{array}} \right.$ The coefficients a ^ij (x, u) are supposed to be VMO functions with respect to x while the term b(x, u, Du) allows controlled growth with respect to the gradient Du and satisfies a sort of sign-condition with respect to u. Our results correct and generalize the announcements in Ragusa (Nonlinear Differ Equ Appl 13:605–617, 2007, Erratum in Nonlinear Differ Equ Appl 15:277–277, 2008).     

Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem

The spectral problem in a bounded domain Ω?Rⁿ is considered for the equation Δu= λu in Ω, ?u=λ?υ/?ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ _j ⁰ } _j=1 ^∞ and {λ _j ^∞ } _j=1 ^∞ , converging respectively to 0 and +∞. It is also established that $$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$ The constants are explicitly calculated.     

Oscillatory Perturbations of Linear Problems at Resonance

This paper is concerned with a study of bounded perturbations of resonant linear problems. It follows from our results that for certain types of bounded domains Ω ? Rⁿ, n ≥ 2, the Dirichlet problem $\matrix{\Delta u+\lambda_{1}u+g(u)=h(x),\ \ \ x\in\Omega\cr \quad\quad\quad\quad\quad\quad u=0,\ \ \ x\in\partial\Omega,}$ has infinitely many positive solutions, in case λ₁ is the principal eigenvalue of ?Δ subject to trivial Dirichlet boundary conditions, g is a nontrivial periodic nonlinearity of zero mean and ∫_03A9h(x)?(x)dx = 0, where ? is an eigenfunction corresponding to λ₁.     

Infinitely many periodic solutions for a class of second-order Hamiltonian systems

In this paper we study the existence of infinitely many periodic solutions for second-order Hamiltonian systems

$$\left\{ {\begin{array}{*{20}c} {\ddot u(t) + A(t)u(t) + \nabla F(t,u(t)) = 0,} \\ {u(0) - u(T) = \dot u(0) - \dot u(T) = 0,} \\ \end{array} } \right.$$

, where F(t, u) is even in u, and ?F(t, u) is of sublinear growth at infinity and satisfies the Ahmad-Lazer-Paul condition.

     

Classical verigin problem as a limit case of verigin problem with surface tension at free boundary

In this psper we consider Verigin problem with surface tension st free     

On the Existence of Multiple Periodic Solutions for the Vector <Emphasis Type="Italic">p</Emphasis>-Laplacian via Critical Point Theory

We study the vector p-Laplacian

A radial symmetry and Liouville theorem for systems involving fractional Laplacian

We investigate the nonnegative solutions of the system involving the fractional Laplacian:

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$

where 0 < α < 1, n > 2, f _i (u), 1 ≤ i ≤ m, are real-valued nonnegative functions of homogeneous degree p _i ≥ 0 and nondecreasing with respect to the independent variables u ₁, u ₂,..., u _m . By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x ₀ if p _i = (n + 2α)/(n ? 2α) for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1 < p _i < (n + 2α)/(n ? 2α) for all 1 ≤ i ≤ m.

     

On a Free Boundary Problem

This paper considers a discontinuous semilinear elliptic problem: \[ -\Delta u=g(u)H(u-\mu )\quad \text{in }\Omega,\qquad u=h\quad \text{on }% \partial \Omega, \] −Δu=g(u)H(u−μ) in Ω, u=h on ∂Ω, where H is the Heaviside function, μ a real parameter and Ω the unit ball in ℝ². We deal with the existence of solutions under suitable conditions on g, h, and μ. It is shown that the free boundary, i.e. the set where u=μ, is sufficiently smooth.     

Regularity of p-harmonic maps from the p-dimensional ball into a sphere

We prove that, forp≥2, all weaklyp-harmonic mapsu=(u ₁,...,u _n ) from thep-dimensional ball into a sphere, i.e. weak solutions of classW ^1,p of the constrained eliptic system $$\begin{gathered} - div(|\nabla u|^{p - 2} \nabla u_i ) = u_i |\nabla u|^p \hfill \\ \sum {(u_i )} ^2 = 1, \hfill \\ \end{gathered} $$ are Hölder continuous. This result is an analogue of an earlier theorem of F. Hélein for the casep=2.     

Solvability of nonlinear systems including (γ, δ)-comparison pairs

Let $\gamma ,\delta \in \mathbb{R}^n $ with $\gamma _j ,\delta _j \in \{ 0,1\} $ . A comparison pair for a system of equations f_i(u₁,…,u_n)=0 (i=1,…,n) is a pair of vectors $v,w \in \mathbb{R}^n ,v \leqslant w$ , such that $$\begin{array}{*{20}c} {\gamma _i f_i (u_1 , \ldots ,u_{i - 1} ,v_i ,u_i + 1, \ldots ,u_n ) \leqslant 0,} \\ {\delta _i f_i (u_1 , \ldots ,u_{i - 1} ,w_i ,u_i + 1, \ldots ,u_n ) \geqslant 0} \\ \end{array} $$ for $\gamma _j u_j \geqslant v_j ,\delta _j u_j \leqslant w_j (j = 1, \ldots ,n)$ . The presence of comparison pairs enables one to essentially weaken the assumptions of the existence theorem. Bibliography: 1 title.     

A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case

We consider here solutions of a nonlinear Neumann elliptic equation Δu +?f (x, u) =?0 in Ω, ?u/?ν =?0 on ?Ω, where Ω is a bounded open smooth domain in ${\mathbb{R}^N, N\geq2}$ and f satisfies super-linear and subcritical growth conditions. We prove that L ^∞?bounds on solutions are equivalent to bounds on their Morse indices.     

De Giorgi type results for elliptic systems

We consider the following elliptic system $${\Delta}u = \nabla H (u) \quad {\rm in}\quad \mathbf{R}^N,$$ where ${u : \mathbf{R}^N \to \mathbf{R}^m}$ and ${H \in C^2(\mathbf{R}^m)}$ , and prove, under various conditions on the nonlinearity H that, at least in low dimensions, a solution ${u=(u_i)_{i=1}^m}$ is necessarily one-dimensional whenever each one of its components u _i is monotone in one direction. Just like in the proofs of the classical De Giorgi’s conjecture in dimension 2 (Ghoussoub-Gui) and in dimension 3 (Ambrosio-Cabré), the key step is a Liouville theorem for linear systems. We also give an extension of a geometric Poincaré inequality to systems and use it to establish De Giorgi type results for stable solutions as well as additional rigidity properties stating that the gradients of the various components of the solutions must be parallel. We introduce and exploit the concept of an orientable system, which seems to be key for dealing with systems of three or more equations. For such systems, the notion of a stable solution in a variational sense coincide with the pointwise (or spectral) concept of stability.     

Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness

In this paper we study viscosity solutions to the system $$\begin{array}{ll} \min \{ -\mathcal{H}u_i(x,t)-\psi _i(x,t),u_i(x,t) - \max_{j \neq i} (-c_{i ,j} (x,t) + u_j (x,t)) \} = 0,\\ u_i(x,T)=g_i (x), \, i \in \{1,\ldots , d \},\end{array}$$ where \({(x,t)\in{\mathbb{R}}^{N} \times [0,T]}\) . Concerning \({{\mathcal{H}}}\) , we assume that \({{\mathcal{H}}={\mathcal{L}}+{\mathcal{I}}}\) where \({{\mathcal{L}}}\) is a linear, possibly degenerate, parabolic operator of second order and \({{\mathcal{I}}}\) is a non-local integro-partial differential operator. A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems when the dynamics of the underlying state variables is described by an N-dimensional Levy process. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the data, i.e., on the operator \({{\mathcal{H}}}\) and on the continuous functions \({\psi_i}\) , c _i,j , and g _i . Using the comparison principle we prove the existence of a unique viscosity solution (u ₁, . . . , u _d ) to the system by Perron’s method. Our main contribution is that we establish existence and uniqueness of viscosity solutions, in the setting of Levy processes and non-local operators, with no sign assumption on the switching costs {c _{i, j} } and allowing c _{i, j} to depend on x as well as t.     

On the selfsimilar solutions of a diffusion convection equation

This paper is concerned with the equation¶¶ div(| ?u| ^p-2?u)+e| ?U| ^q+bx?U+aU=0, for  x ? \mathbbR^N div(| \nabla u| ^{p-2}\nabla u)+\varepsilon \left| \nabla U\right| ^q+\beta x\nabla U+\alpha U=0,{\rm \ for}\;x\in \mathbb{R}^N ¶¶ where $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 and a,b, m \alpha ,\beta, \mu are positive parameters. We study the existence, uniqueness of radial solutions u(r). Also, qualitative behavior of u(r) are presented.     

An infinite algebraic system in the irregular case

We obtain sufficient conditions for the nontrivial solvability of systems of the form $$ \phi _i = b_i + \lambda _i \sum\limits_{j = 0}^\infty {a_{i - j} \phi _j ,i \in \mathbb{Z}_ + \underline{\underline {def}} \{ 0,1,2...,n,...\} ,} $$ and of the corresponding homogeneous systems. It is assumed that the sequences b = (b ₀, b ₁, b ₂, …) and λ = (λ ₀, λ ₁, λ ₂, …) and the Toeplitz matrix A = (a _i?j ) satisfy the conditions $$ \begin{gathered} a_j \geqslant 0,j \in \mathbb{Z},\sum\limits_{j = - \infty }^\infty {a_j = 1,} \sum\limits_{j = - \infty }^\infty {|j|a_j < \infty ,\sum\limits_{j = - \infty }^\infty {ja_j < 0,} } \hfill \\ b_j \geqslant 0,j \in \mathbb{Z},\sum\limits_{j = 0}^\infty {b_j = \infty ,} 1 \leqslant \lambda _i \leqslant \left( {\sum\limits_{j = - \infty }^i {a_j } } \right)^{ - 1} ,i \in \mathbb{Z}_ + . \hfill \\ \end{gathered} $$ . Under these conditions, we construct bounded solutions of homogeneous and inhomogeneous systems of the form indicated above.     

Bound states for a coupled Schrödinger system

We consider the existence of bound states for the coupled elliptic system