A random walk for the solution sought: Remarks on the difference scheme approach to nonlinear semigroups and evolution operators

In [2], Crandall and Evans show existence of mild solution to an abstract Cauchy Problem: u′(t)+Au(t)∋f(t), 0≤t≤T, u(0)=x₀, where A is an accretive operator in a general Banach space X and f ε L¹(0,T;X). Their method involves proving convergence in the L^∞-norm of a sequence of step function approximations α_n(σ, τ) to the solution of a first order partial differential equation. We consider a more general Cauchy Problem and show a.e. existence of mild solution by proving convergence of the step functions α_n(σ, τ) in the L¹-norm. Fundamental to the proof is a nonhomogeneous random walk in the plane.     

Stein domains in complex surfaces

Let S be a closed connected real surface and π: S→X a smooth embedding or immersion of S into a complex surface X. We denote by I(π) (resp. by I_±(π) if S is oriented) the number of complex points of π (S)∪X counted with algebraic multiplicities. Assuming that I(π)≤0 (resp. I_±(π)≤0 if S is oriented) we prove that π can be C⁰ approximated by an isotopic immersion π₁: S→X whose image has a basis of open Stein neighborhood in X which are homotopy equivalents to π₁ (S). We obtain precise results for surfaces in and find an immersed symplectic sphere in with a Stein neighborhood.     

Linear series onk-gonal curves

Here we prove the following result. Theorem 1.1.Let X be an integral projective curve of arithmetic genus g and k≧ ≧4 an integer. Assume the existence of L ∈ Pic^k (X) with h ⁰ (X, L)=2 and L spanned. Fix a rank 1 torsion free sheaf M on X with h ⁰(X,M)=r+1≧2, h¹ (X, M)≧2 and M spanned by its global sections. Set d≔deg(M) and s≔max {n≧0:h ⁰ (X, M ⊗(L^*)^⊗n)>0}. Then one of the following cases occur:

A boundary-value problem for the biharmonic equation and the iterated Laplacian in a 3D-domain with an edge

Let Ω be a domain with piecewise smooth boundary. In general, it is impossible to obtain a generalized solution u ∈ W ₂ ² (Ω) of the equation Δ _x ² u = f with the boundary conditions u = Δ_xu = 0 by solving iteratively a system of two Poisson equations under homogeneous Dirichlet conditions. Such a system is obtained by setting v = −Δu. In the two-dimensional case, this fact is known as the Sapongyan paradox in the theory of simply supported polygonal plates. In the present paper, the three-dimensional problem is investigated for a domain with a smooth edge Γ. If the variable opening angle α ∈ C^∞(Γ) is less than π everywhere on the edge, then the boundary-value problem for the biharmonic equation is equivalent to the iterated Dirichlet problem, and its solution u inherits the positivity preserving property from these problems. In the case α ∈ (π 2π), the procedure of solving the two Dirichlet problems must be modified by permitting infinite-dimensional kernel and co-kernel of the operators and determining the solution u ∈ W ₂ ² (Ω) by inverting a certain integral operator on the contour Γ. If α(s) ∈ (3π/2,2π) for a point s ∈ Γ, then there exists a nonnegative function f ∈ L₂(Ω) for which the solution u changes sign inside the domain Ω. In the case of crack (α = 2π everywhere on Γ), one needs to introduce a special scale of weighted function spaces. In this case, the positivity preserving property fails. In some geometrical situations, the problems on well-posedness for the boundary-value problem for the biharmonic equation and the positivity property remain open. Bibliography: 46 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 153–198.     

Representation results for law invariant time consistent functions

We show that the only dynamic risk measure which is law invariant, time consistent and relevant is the entropic one. Moreover, a real valued function c on L ^∞(a, b) is normalized, strictly monotone, continuous, law invariant, time consistent and has the Fatou property if and only if it is of the form ${c(X)=u^{-1} \circ\mathbb {E}[u(X)]}We show that the only dynamic risk measure which is law invariant, time consistent and relevant is the entropic one. Moreover, a real valued function c on L ^∞(a, b) is normalized, strictly monotone, continuous, law invariant, time consistent and has the Fatou property if and only if it is of the form c(X)=u^-1 °\mathbb E[u(X)]{c(X)=u^{-1} \circ\mathbb {E}[u(X)]} , where u:(a, b) ? \mathbb R{u:(a, b) \to {\mathbb R}} is a strictly increasing, continuous function. The proofs rely on a discrete version of the Skorohod embedding theorem.     

Solvability of nonlinear equations in a cone of a banach space

The solvability conditions for the equation Tu+F(u)=0 are found in the case where the operator [T+F′(u)]⁻¹ exists only for u∈K, where K is a cone in a Banach space X. An application concerning the solvability of boundary-value problems for systems of second-order differential equations is provided. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 225–230. Translated by L. Yu. Kolotilina.     

Evolution equations with a free boundary condition

In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂₁,u −Δu = Γ(u)(∇u, ∇u) [tT T_{u u}N, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥_u T_u(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = u_o(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M. Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝⁿ, in which the nonlinear equation reduces to ∂_tu-Δu = 0. We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than K_N <-0 and Σ is totally geodesic in N.     

The first boundary-value problem for a singular nonlinear ordinary differential equation of fourth order

The solvability of the boundary-value problem

The Chow ring of a singular surface

We construct the Chow ringCH*(X) =CH ⁰ (X)⊕CH ¹ (X)⊕CH ² (X) of a reduced, quasi-projective surfaceX, together with Chern class mapsc _i :K ₀ (X) → CH ⁱ (X), with the usual properties. As a consequence, we show that the cycle mapCH ² (X)→ F ₀ K ₀ (X) is an isomorphism. Our treatment is greatly influenced by an unpublished 1983 preprint of Levine’s, but is much simpler, since we deal only with surfaces.     

A local limit theorem for random walks conditioned to stay positive

We consider a real random walk S_n=X₁+...+X_n attracted (without centering) to the normal law: this means that for a suitable norming sequence a_n we have the weak convergence S_n/a_n⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S₁>0,...,S_n>0) and let S_n⁺ denote the random variable S_n conditioned on : it is known that S_n⁺/a_n ↠ ϕ⁺(x) dx, where ϕ⁺(x):=x exp (−x²/2)1_(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X₁ has an absolutely continuous law: in this case the uniform convergence of the density of S_n⁺/a_n towards ϕ⁺(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.     

Homogenization of quasi-linear equations with natural growth terms

In this paper we deal with the limit behaviour of the bounded solutions u_ε of quasi-linear equations of the form of Ω with Dirichlet boundary conditions on σΩ. The map a=a(x,ϕ) is periodic in x, monotone in ϕ, and satisfies suitable coerciveness and growth conditions. The function H=H(x,s,ϕ) is assumed to be periodic in x, continuous in [s,ϕ] and to grow at most like |ξ|^p. Under these assumptions on a and H we prove that there exists a function H⁰=H⁰(s,ϕ) with the same behaviour of H, such that, up to a subsequence, (u_ε) converges to a solution u of the homogenized problem -div(b(Du)) + γ|u|^p-2u = H⁰(u,Du) + h(x) on Ω, where b depends only on a and has analogous qualitative properties.     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type

We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) nonlocally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differential-geometric objects related to this system, namely, the matrix (affinor) V_jⁱ(u) of this system of hydrodynamic type, the metrics g ₁ ^ij(u) and g ₂ ^ij(u), the affinor υ_jⁱ(u) = g ₁ ^is(u)g _2,sj(u), and also the affinors (w _1,n)_jⁱ(u) and (w _2,n)_jⁱ(u) of the nonsingular nonlocal bi-Hamiltonian structure of this system, are diagonal in these special “diagonalizing” local coordinates (Riemann invariants of the system). The proof is a natural corollary of the general results of our previously developed theories of compatible metrics and of nonlocal bi-Hamiltonian structures; we briefly review the necessary notions and results in those two theories.     

Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

We present existence principles for the nonlocal boundary-value problem (φ(u^(p−1)))′=g(t,u,...,u^(p−1), α_k(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α _k: C ^p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)ⁿ(φ(u⁽²ⁿ⁻⁾))′=f(t,u,...,u⁽²ⁿ⁻¹⁾), u^(2k)(0)=0, α_ku^(2k)(T)+b_ku^(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.     

Homogenization of the parabolic cauchy problem in the Sobolev class H<Superscript>1</Superscript>(R<Superscript>d</Superscript>)

Homogenization in the small period limit for the solution ue of the Cauchy problem for a parabolic equation in R^d is studied. The coefficients are assumed to be periodic in R^d with respect to the lattice ɛG. As ɛ → 0, the solution u ɛ converges in L2(R^d) to the solution u0 of the effective problem with constant coefficients. The solution u ɛis approximated in the norm of the Sobolev space H ¹(R^d) with error O( ɛ); this approximation is uniform with respect to the L2-norm of the initial data and contains a corrector term of order ɛ. The dependence of the constant in the error estimate on time t is given. Also, an approximation in H ¹(R^d) for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained.     

A Robin boundary problem with Hardy potential and critical nonlinearities

Let Ω be a bounded domain with a smooth C ² boundary in ℝⁿ (n ≥ 3), 0 ∈ , and υ denote the unit outward normal to ∂Ω. In this paper, we are concerned with the following class of boundary value problems:

Growth properties of plurisubharmonic functions related to Fourier-Laplace transforms

The purpose of the paper is to study the behavior at infinity of Fourier-Laplace transforms of distributions or more generally plurisubharmonic functions u in Cⁿ with bounds of the form

Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires

Résumé Soit X un processus gaussien stationnaire non dérivable. Nous étudions le nombre de passages en zéro du processus régularisé par convolution. Sous des hypothèses peu restrictives sur X, cette variable convenablement normalisée, converge au sens de L ² quand la taille du filtre tend vers zéro. Lorsque X admet un temps local continu, la limite obtenue est le temps local.

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Summary Let {X(t)} be a stationary non differentiable Gaussian process and let ϕ_ɛ(u=ɛ⁻¹ ϕ(u/ɛ) be an approximate identity. Setting X _ɛ(t)=X*ϕ_ɛ(t) and letting N _ɛ(T) be the number of zeros of X _ɛ in the interval [0, T] it is shown that under weak technical conditions there are constants C(ɛ) so that C(ɛ) N _ɛ(T) converges in L ² as ɛ→0. When X admits a continuous local time, the limit is the local time L(0, T) at zero of X(t).

     

Sviluppo delle funzioni implicitamente definite da un sistema di equazioni nel campo reale

Sunto Per la funzione reale del punto P(x, y): F[ξ(P), η(P)], ove u=ξ(P), v=η(P) sono implicitamente definite nel campo reale dal sistema delle due equazioni u−x+ϕ(u, v)=0, v−y+φ(u, v)=0 si dà uno sviluppo che estende quello ottenuto dalLevi-Civita per la funzione F[y(x)], ove y(x) è definita dalla equazione y−x+ϕ(y)=0. Ad Antonio Signorini nel suo 70^mo compleanno.