Comportement asymptotique des solutions d'un système elliptique conservatif non linéaire

Let M be a compact riemannian manifold; in a previous article we show that every non- negative solution of u_tt + Δ_g u=f(u) on M ×R ⁺, satisfying Dirichlet or Neumann boundary conditions, converges to a (stationary) solution Φ of Δ_g Φ=f(Φ) with exponential decay of ∥u - Φ∥c²(M), if we assume that f behaves like r? r^p - λr. We extend this result to a system in the following form $$\left\{ {\begin{array}{*{20}c} {u_{tt} + \Delta _g u + \alpha u - G_x (u,\upsilon ) = 0,} \\ {u_{tt} + \Delta _g \upsilon + \beta \upsilon - G_x (u,\upsilon ) = 0,} \\ \end{array} } \right.$$ . where G satisfies some growth and convexity properties.     

Generalized <Emphasis Type="Italic">n</Emphasis>-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Let n ≥ 2 and let Ω ? ? ⁿ be an open set. We prove the boundedness of weak solutions to the problem

$$u \in W_0^1 L^\Phi \left( \Omega \right) and - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}}{{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u}{{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega ,$$

where ? is a Young function such that the space W ₀ ¹ L ^Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L ^Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω = ? ⁿ .

     

Critical sobolev exponent problem in ℝ n (n ≥ 4) with neumann boundary condition

In this paper we study the existence and non existence of positive solution for the critical Sobolev exponent problem ? Δu =u(n + 2)/(n ? 2) +λα(x)u) in Ω \(\frac{{\partial u}}{{\partial v}} = 0 on \partial B\) , where Ω is a bounded domain in ? ⁿ (n ≥ 4).     

On solvability of Dirichlet problem for second order elliptic equation

The paper gives some solvability conditions of the Dirichlet problem for the second order elliptic equation $$ - div(A(x)\nabla u) + (\bar b(x),\nabla u) - div(\bar c(x)u) + d(x)u = f(x) - divF(x),x \in Q,u|_{\partial Q} = u_0 \in L_2 (\partial Q) $$ in bounded domain Q ? R _n (n ≥ 2) with smooth boundary ?Q ∈ C ₁. In particular, it is proved that if the homogeneous problem has only the trivial solution, then for any u ₀ ∈L ₂(?Q) and f, F from the corresponding functional spaces the solution of the non-homogeneous problem exists, from Gushchin’s space $ C_{n - 1} (\bar Q) $ and the following inequality is true: $$ \begin{gathered} \left\| u \right\|_{C_{n - 1} (\bar Q)}^2 + \mathop \smallint \limits_Q r\left| {\nabla u} \right|^2 dx \leqslant \hfill \\ \leqslant C\left( {\left\| {u_0 } \right\|_{L_2 (\partial Q)}^2 + \mathop \smallint \limits_Q r^3 (1 + |\ln r|)^{3/2} f^2 dx + \mathop \smallint \limits_Q r(1 + |\ln r|)^{3/2} |F|^2 dx} \right) \hfill \\ \end{gathered} $$ where r(x) is the distance from a point x ∈ Q to the boundary ?Q and the constant C does not depend on u ₀, f and F.     

On continuous solutions of a problem of R.Schilling

Abstract. It is proved that if $$ q \ \in \lbrace({\sqrt 3}-1)/2,(3-{\sqrt 5})/2,{\sqrt 2}-1,({\sqrt 5}-1)/2\rbrace $$ then the zero function is the only solution ?: ? → ? of (1) satisfying (2) and right-hand-side or left-hand-side continuous at each point of the interval (?q/(1 ? q), ?q/(1 ? g) + δ) or of the interval (q/(1 ? q) ? δ, q/(1 ? q)) with some δ > 0.     

Generalized n-Laplacian: Semilinear Neumann problem with the critical growth

Let Ω ? ? ⁿ , n ? 2, be a bounded connected domain of the class C ^1,θ for some θ ∈ (0, 1]. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem $$\begin{gathered} u \in W^1 L^\Phi \left( \Omega \right), - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}} {{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u} {{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega , \hfill \\ \frac{{\partial u}} {{\partial n}} = 0 on \partial \Omega , \hfill \\ \end{gathered}$$ where Φ is a Young function such that the space W ¹ L ^Φ(Ω) is embedded into exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V (x) is a continuous potential, h ∈ (L ^Φ(Ω))* is a nontrivial continuous function, µ ? 0 is a small parameter and n denotes the outward unit normal to ?Ω.     

Summation of weakly dependent sequences by Cesàro methods

Пусть (X, A, u) — пространст во с конечной мерой, (ξ_k) ₁ ^∞ — последовательност ь функций, \(\xi _k \varepsilon L_{2r} (X), r > 1, \int\limits_X {\xi _k d\mu = 0} \) . Изучаются условия, п ри которых справедли вgа - у. з. б.ч., т. e. (ξ _k) суммируется к ну лю почти всюду методо м (С, а),а > 0. Приведем два резу льтата. 1) Если (ξ _k) — слабо мульт ипликативная систем а (в частности, мартингал-разности или независимая сист ема), то условие $$\mathop \sum \limits_1^\infty \mathop {\smallint }\limits_X \left| {\xi _k } \right|^{2r} d\mu \cdot c_r (k,\alpha )< \infty $$ влечетβ - у.з.б.ч. Здесьc _r(k,α)=k ^-2rα при 0<α<(r+1)/2r, c_r=k^?(r+1) In3r-1 k приа=(r+1)/2r, с_r=k^?(r+1) при а >(r+1)/2r. 2) Если (ξ _k) независимы,Mξ _k=0, (r+1)/2r<α=1, то условия $$\mathop \sum \limits_{k = 1}^\infty \frac{{(M\xi _k^2 )^r }}{{k^{r + 1} }}< \infty ,\mathop \sum \limits_{k = 1}^\infty \frac{{M|\xi _k |^{2r} }}{{k^{2r\alpha } }}< \infty $$ влекут за собой а - у. з. б. ч.     

Sufficient conditions that the minimum and maximum of partial differential operators should coincide and that their spectra should be discrete

An expression of the form $$l(u) = ( - 1)^m \sum\nolimits_{j = 1}^m {D_j^{2m} } u + [q(x)] + ir(x)]u$$ is considered. Sufficient conditions are found such that the minimum operator, formally conjugate tol(u), generated by the expression and the maximum operator generated by the expressionl(u) in ?₂ (E_n) should coincide. It is proved that if q(x)→∞ or q(x)+ r(x)→∞, ¦x¦→∞, then the operator generated byl(u) in ?₂ (E_n) has a discrete spectrum.     

On almost good triples of vertices in edge regular graphs

Consider a connected edge regular graph Γ with parameters (v, k, λ) and put b ₁ = k?λ?1. A triple (u, w, z) of vertices is called (almost) good whenever d(u, w) = d(u, z) = 2 and µ(u, w)+µ(u, z) ≤ 2k ? 4b ₁ + 3 (and µ(u, w) + µ(u, z) = 2k ? 4b ₁ + 4). If k = 3b ₁ + γ with γ ≥ ?2, a triple (u, w, z) is almost good, and Δ = [u] ∩ [w] ∩ [z] then: either |Δ| ≤ 2; or Δ is a 3-clique and Γ is a Clebsch graph; or Δ is a 3-clique, k = 16, b ₁ = 6, and v = 31; or Δ is a 4-clique and Γ is a Schläfli graph.     

Boundary value problems for complete quasi-parabolic differential equations of odd order with variable domains of operators

We prove the well-posed solvability in the strong sense of the boundary value Problems

$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$

where the unbounded operators A _s (t), s > 0, in a Hilbert space H have domains D(A _s (t)) depending on t, are subordinate to the powers A ^1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A ₀(t) and the symmetric operators A _s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that

$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$

, where the smoothing operators are defined by

$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$

.

     

Hooked extended Langford sequences of small and large defects

It is shown that for m = 2d +5, 2d+6, 2d+7 and d ≥ 1, the set {1, …, 2m + 1} ? {k} can be partitioned into differences d, d + 1, …, d + m ? 1 whenever (m, k) ≡ (0, 1), (1, d), (2, 0), (3, d+1) (mod (4, 2)) and 1 ≤ k ≤ 2m+1. It is also shown that for m = 2d + 5, 2d + 6, 2d + 7, and d ≥ 1, the set {1, …, 2m + 2} ? {k, 2m + 1} can be partitioned into differences d, d + 1, … …, d + m ? 1 whenever (m, k) ≡ (0, 0), (1, d + 1), (2, 1), (3, d) (mod (4, 2)) and k ≥ m + 2. These partitions are used to show that if m ≥ 8d + 3, then the set {1, … …, 2m+2}?{k, 2m+1} can be partitioned into differences d, d+1, …, d+m?1 whenever (m, k) ≡ (0, 0), (1, d+1), (2, 1), (3, d) (mod (4, 2)). A list of values m, d that are open for the existence of these partitions (which are equivalent to the existence of Langford and hooked Langford sequences) is given in the conclusion.     

Examples of equations of Navier-Stokes type not strongly solvable in the large

In the space of functions with values in Hilbert space, we consider the Cauchy problem u′ _t + Au + B(u, u) = f(t), u(0) = 0, 0 ≤ t ≤ T. We construct examples of a self-adjoint operator A ≥ E and a bilinear transformation B satisfying the condition 〈B(u, v), v〉 = 0 such that the Cauchy problem is not strongly solvable.     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

Inverse cyclotomic polynomials

Let Ψn(x) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψn(x)=(xn−1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small in absolute value, e.g. for n<561 all coefficients of Ψn(x) are ?1 in absolute value. We establish various properties of the coefficients of Ψn(x), especially focusing on the easiest non-trivial case where n is composed of 3 distinct odd primes.     

GLOBAL EXISTENCE OF THE SOLUTION OF NONLINEAR SCHRDINGER EQUATIONS IN EXTERIOR DOMAINS

In this paper,we discuss the problem for the nonlinear Schr(?)dinger equation(?)where Ω is the exterior domain of a compact set in B~n,a_j(u)=O(|u|),b_j(u)=O(|u|)(1≤j≤n),c(u)=O(|u|~2)near u=0.If n≥5,some Sobolev norm of u_0(x)is sufficiently small,under suitableassumptions on smoothnessand and compatibility and the shape of Ω,we get that the problem has aunique global solution u(t,x),with the decay estimate‖u(t,·)‖_(L(?)(Ω))=O(t~(-n/4)),‖u(t,·)‖_(L~4(Ω))=O(t~(-n/4)),t→+∞.     

Series inversion of some convolution transforms

We study degeneration for ? → + 0 of the two-point boundary value problems

$\text{τ?}{}^{\mathrm{\pm }}\text{u := ?((au′)′ + bu′ + cu) ± xu′ ? κu = h, u(±1) = A ± B}$

, and convergence of the operators T_?⁺ and T_?^? on $\text{L}$²(?1, 1) connected with them, T_?^±u := τ_?^±u for all

$\text{u?D(T}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{, D(T}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{) := {u ? L}{}^2\text{(?1, 1) ∣ u″ ? L}{}^2\text{(?1, 1) \&; u(?1) = u(1) = O}, T}{}_0{}^{\mathrm{+}}\text{u: = xu′}$

for all

$\text{u?D(T}{}_{\mathrm{O}}{}^{\mathrm{+}}\text{), D(T}{}_{\mathrm{O}}{}^{\mathrm{+}}\text{) := {u ? L}{}^2\text{(?1, 1) ∣ xu′ ? L}{}^2\text{(?1, 1) \&; u(?1) = u(1) = O}}$

. Here ? is a small positive parameter, λ a complex “spectral” parameter; a, b and c are real $\text{b}$^∞-functions, a(x) ? γ > 0 for all x? [?1, 1] and h is a sufficiently smooth complex function. We prove that the limits of the eigenvalues of T_?⁺ and of T_?^? are the negative and nonpositive integers respectively by comparison of the general case to the special case in which a  1 and b  c  0 and in which we can compute the limits exactly. We show that (T_?⁺ ? λ)^?1 converges for ? → +0 strongly to (T₀⁺ ? λ)^?1 if $\text{R e λ > ?}\text{1}\text{2}$. In an analogous way, we define the operator T_?⁺, n (n ? $\text{N}$ in the Sobolev space H₀^?n(? 1, 1) as a restriction of τ_?⁺ and prove strong convergence of (T⁺_?,n ? λ)^?1 for ? → +0 in this space of distributions if $\text{R e λ > ?n ?}\text{1}\text{2}$. With aid of the maximum principle we infer from this that, if h?$\text{C}$¹, the solution of τ_?⁺u ? λu = h, u(±1) = A ± B converges for ? → +0 uniformly on [?1, ? ?] ∪ [?, 1] to the solution of xu′ ? λu = h, u(±1) = A ± B for each p > 0 and for each λ ? $\text{C}$ if ? ?$\text{N}$.Finally we prove by duality that the solution of τ_?^?u ? λu = h converges to a definite solution of the reduced equation uniformly on each compact subset of (?1, 0) ∪ (0, 1) if h is sufficiently smooth and if 1 ? ?$\text{N}$.     

On the existence and smoothness of a generalized solution of a nonlinear differential equation with degeneration

Let Ω be an arbitrary open set in R _n , and let σ(x) and g _i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W _p ^r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g ₁(x), g ₂(x), ..., g _n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W _p ^r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.     

Geometry and entropy of generalized rotation sets

For a continuous map f on a compact metric space we study the geometry and entropy of the generalized rotation set Rot(Φ). Here Φ = (?₁, ..., ? _m ) is a m-dimensional continuous potential and Rot(Φ) is the set of all µ-integrals of Φ and µ runs over all f-invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of ? ^m . We study the question if every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set K in ? ^m a potential Φ = Φ(K) with Rot(Φ) = K. Next, we study the relation between Rot(Φ) and the set of all statistical limits Rot _Pt (Φ). We show that in general these sets differ but also provide criteria that guarantee Rot(Φ) = Rot _Pt (Φ). Finally, we study the entropy function w ? H(w),w ∈ Rot(Φ). We establish a variational principle for the entropy function and show that for certain non-uniformly hyperbolic systems H(w) is determined by the growth rate of those hyperbolic periodic orbits whose Φ-integrals are close to w. We also show that for systems with strong thermodynamic properties (sub-shifts of finite type, hyperbolic systems and expansive homeomorphisms with specification, etc.) the entropy function w ? H(w) is real-analytic in the interior of the rotation set.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

On solutions to coupled multiparameter nonlinear Sturm- Liouville boundary value problems whose state components have a specified nodal structure

In preceding articles ([3] and [5]), we began an examination of the structure of the solution set to the two-parameter system \(\matrix{\qquad\qquad-(p_{1}(x)u^{\prime})^{\prime}+q_1(x)u= \lambda u+f(u,v)u\cr \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad in (a,b)\cr \qquad(*)\ \ -(p_{2}(x)v^{\prime})^\prime+ q_{2}(x)v=\mu v+g(u,v)v\cr \qquad\qquad u(a)=u(b)=0=v(a)=v(b).}\) In this article, we treat the case left uncovered in our previous analysis; namely, we assume f (s,0) = 0 and g(0,t) = 0 for all s, t ∈ ?. In this situation, solutions to (*) of the form (λ, μ, u, 0) or (λ, μ,0,v lie in linear subspaces of ?.² dx (C01 [a, b] 2. As such, they are neither locally expressable as functions of (λ, μ) nor are à priori bounded in terms of (λ, μ), as was crucial to the analysis in [3] and [5]. Nevertheless, we demonstrate that solutions to (*) of the form (λ, μ, u, v) with u having n ? 1 simple zeros in (a, b) and v having m ? 1 simple zeros in (a, b), where n and m are positive integers, arise as global secondary bifurcations from solutions of the form (λ,μ,u,0) with u having n ? 1 simple zeros in (a, b) and from solutions of the form (λ, μ, o, v) with v having m ? 1 simple zeros in (a, b). Moreover, we establish that solutions to (*) of the form (λ,μ,u,v) with u having n ? 1 simple zeros in (a, b) and v having m ? 1 simple zeros in (a, b) serve as a link between solutions of the form (λ, μ, u, 0) with u having n ? 1 simple zeros in (a,b) and solutions of the form (λ, μ, 0, v) with v having m ? 1 simple zeros in (a, b). The analysis in this article when combined with that in [3] and [5] provides a fairly comprehensive examination of the structure of the solution set to (*).