Gradient Estimates of q-Harmonic Functions of Fractional Schrödinger Operator

We study gradient estimates of q-harmonic functions u of the fractional Schrödinger operator Δ ^α/2?+?q, α?∈?(0, 1] in bounded domains D???? ^d . For nonnegative u we show that if q is Hölder continuous of order η?>?1???α then $\nabla u(x)$ exists for any x?∈?D and $|\nabla u(x)| \le c u(x)/ ({\rm dist}(x,\partial D) \wedge 1)$ . The exponent 1???α is critical i.e. when q is only 1???α Hölder continuous $\nabla u(x)$ may not exist. The above gradient estimates are well known for α?∈?(1, 2] under the assumption that q belongs to the Kato class $\mathcal{J}^{\alpha - 1}$ . The case α?∈?(0, 1] is different. To obtain results for α?∈?(0, 1] we use probabilistic methods. As a corollary, we obtain for α?∈?(0, 1) that a weak solution of Δ ^α/2 u?+?q u?=?0 is in fact a strong solution.     

General páley problem

In the class of functionsu of finite lower order subharmonic in ? ^p+2,p ∈ ? we establish an exact upper bound for $$\mathop {\lim }\limits_{r \to \infty } \inf \frac{{m_q (r,u^ + )}}{{T(r,u)}}, 1< q \le \infty ,$$ whereT(r, u) is a Nevanlinna characteristic of the functionu andm _q (r, u ⁺) is the integralq-mean of the functionu ⁺,u ⁺ = max(u,0), on the sphere of radiusr.     

On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? ^N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u ^p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v ^p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.     

Eigenvalue problem for finite difference equations with p-Laplacian

In this paper, we consider a discrete four-point boundary value problem $$\triangle\bigl(\phi_p\bigl(\triangle u(k-1)\bigr)\bigr)+ \lambda e(k)f\bigl(u(k)\bigr)=0,\quad k\in N(1,T),$$ subject to boundary conditions $$\triangle u(0)-\alpha u(l_{1})=0,\qquad\triangle u(T)+\beta u(l_{2})=0,$$ by a simple application of a fixed point theorem. If e(k),f(u(k)) are nonnegative, the solutions of the above problem may not be nonnegative, this is the main difficulty for us to study positive solution of this problem. In this paper, we give restrictive conditions ??l ₁??1, ??(T+1?l ₂)??1 to guarantee the solutions of this problem are nonnegative, if it has, under the conditions e(k),f(u(k)) are nonnegative. We first construct a new operator equation which is equivalent to the problem and provide sufficient conditions for the nonexistence and existence of at least one or two positive solutions. In doing so, the usual restrictions $f_{0}=\lim_{u\rightarrow 0^{+}}\frac{f(u)}{\phi_{p}(u)}$ and $f_{\infty}=\lim_{u\rightarrow\infty}\frac{f(u)}{\phi_{p}(u)}$ exist are removed.     

Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions

In a simply connected two dimensional domain Ω, we consider Ginzburg-Landau minimizers u with zero degree Dirichlet boundary condition ${g \in H^{1/2}(\partial \Omega; \mathbb{S}^1)}$ . We prove uniqueness of u whenever either the energy or the Ginzburg-Landau parameter are small. This generalizes a result of Ye and Zhou requiring smoothness of g. We also obtain uniqueness when Ω is multiply connected and the degrees of the vortexless minimizer u are prescribed on the components of the boundary, generalizing a result of Golovaty and Berlyand for annular domains. The proofs rely on new global estimates connecting the variation of |u| to the Ginzburg-Landau energy of u. These estimates replace the usual global pointwise estimates satisfied by ${\nabla u}$ when g is smooth, and apply to fairly general potentials. In a related direction, we establish new uniqueness results for critical points of the Ginzburg-Landau energy.     

Isolated boundary singularities of semilinear elliptic equations

Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u ^q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 0. We also investigate the case ${q= \frac{N+1}{N-1}}$ . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.     

Convergent approximations in parabolic variational inequalities II: Hamilton-jacobi inequalities

In this paper we consider two-sided parabolic inequalities of the form (li) $$\psi _1 \leqslant u \leqslant \psi _2 , in{\mathbf{ }}Q;$$ (lii) $$\left[ { - \frac{{\partial u}}{{\partial t}} + A(t)u + H(x,t,u,Du)} \right]e \geqslant 0, in{\mathbf{ }}Q,$$ for alle in the convex support cone of the solution given by $$K(u) = \left\{ {\lambda (\upsilon - u):\psi _1 \leqslant \upsilon \leqslant \psi _2 ,\lambda > 0} \right\}{\mathbf{ }};$$ (liii) $$\left. {\frac{{\partial u}}{{\partial v}}} \right|_\Sigma = 0, u( \cdot ,T) = \bar u$$ where $$Q = \Omega \times (0,T), \sum = \partial \Omega \times (0,T).$$ Such inequalities arise in the characterization of saddle-point payoffsu in two person differential games with stopping times as strategies. In this case,H is the Hamiltonian in the formulation. A numerical scheme for approximatingu is obtained by the continuous time, piecewise linear, Galerkin approximation of a so-called penalized equation. A rate of convergence tou of orderO(h ^1/2) is demonstrated in theL ²(0,T; H ¹(Ω)) norm, whereh is the maximum diameter of a given triangulation.     

Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

For the equation $$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$ whereh ₀ ^o =0,h ₀ ¹ =0 (t) ≡ 0,h _j ^o = const > 0,h ₁ ^j (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation $$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$ connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.     

A counterexample to the strong subadditivity of extremal plurisubharmonic functions

A simple example is given which shows that one way have $$h_E (z^0 ) + h_F (z^0 ) > h_{E \cup F} (z^0 ) + h_{E \cap F} (z^0 )$$ for some pointz ⁰∈ω, where $$h_E (z) = \sup \{ u(z):u \in PSH (\Omega ),u \leqslant 0 on E,u \leqslant 1 in \Omega \} ,z \in \Omega ,$$ is the extremal function often studied in complex analysis.     

Convergence of approximate solutions to an elliptic–parabolic equation without the structure condition

We study the Cauchy–Dirichlet problem for the elliptic–parabolic equation $$b(u)_t + {\rm div} F(u) - \Delta u = f$$ in a bounded domain. We do not assume the structure condition $$b(z) = b(\hat z) \Rightarrow F(z) = F(\hat z).$$ Our main goal is to investigate the problem of continuous dependence of the solutions on the data of the problem and the question of convergence of discretization methods. As in the work of Ammar and Wittbold (Proc R Soc Edinb 133A(3):477–496, 2003) where existence was established, monotonicity and penalization are the main tools of our study. In the case of a Lipschitz continuous flux F, we justify the uniqueness of u (the uniqueness of b(u) is well-known) and prove the continuous dependence in L ¹ for the case of strongly convergent finite energy data. We also prove convergence of the ${\varepsilon}$ -discretized solutions used in the semigroup approach to the problem; and we prove convergence of a monotone time-implicit finite volume scheme. In the case of a merely continuous flux F, we show that the problem admits a maximal and a minimal solution.     

Decay of solutions of the wave equation with a local degenerate dissipation

We derive a precise decay estimate of the solutions to the initial-boundary value problem for the wave equation with a dissipation:u _tt ? Δu+a(x)u _t =0 in Ω × [0, ∞) with the boundary conditionu/?Ω, wherea(x) is a nonnegative function on $\bar \Omega $ satisfying $$a(x) > a.e. x \in \omega and\smallint _\omega \frac{1}{{a(x)^P }}dx< \infty for some 0< p< 1$$ for an open set $\omega \subset \bar \Omega $ including a part of ?Ω with a specific property. The result is applied to prove a global existence and decay of smooth solutions for a semilinear wave equation with such a weak dissipation.     

L 1 decay properties for a semilinear parabolic system

This article is concerned with the decay property in theL ¹ norm ast»∞ of the nonnegative solutions of the initial value problem in ? ⁿ $\left\{ {\begin{array}{*{20}c} {u_t = \Delta u + \mu |\nabla \upsilon |^q } \\ {\upsilon _t = \Delta \upsilon + \upsilon |\nabla \upsilon |^p } \\ \end{array} } \right.$ for different values of the parametersp, q≥1 and when μ, ν<0. If $pq > \frac{{\inf \left( {p,q} \right)}}{{n + 1}} + \left( {n + 2} \right)/\left( {n + 1} \right)$ then lim _t→∞∥u(t)+v(t)∥₁>0 and when $pq< \frac{{\inf \left( {p,q} \right)}}{{n + 1}} + \left( {n + 2} \right)/\left( {n + 1} \right)$ then lim _t→∞∥u(t)+v(t)∥₁>0.     

On the stationary Navier–Stokes flows around a rotating body

Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body $ \mathcal{K} = \mathbb{R}^3 \, \backslash \, {\Omega}$ which is also moving in the direction of the axis of rotation. We assume that the translational and angular velocities U, ω are constant and the external force is given by f = div F. Then the motion is described by a variant of the stationary Navier–Stokes equations on the exterior domain Ω for the unknown velocity u and pressure p, with U, ω, F being the data. We first prove the existence of at least one solution (u, p) satisfying ${\nabla u, p \in L_{3/2, \infty} (\Omega)}$ and ${u \in L_3, \infty (\Omega)}$ under the smallness condition on ${|U| + |\omega| + ||F||_{L_{3/2, \infty} (\Omega)}}$ . Then the uniqueness is shown for solutions (u, p) satisfying ${\nabla u, p \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}$ and ${u \in L_{3, \infty} (\Omega) \cap L_{q*, r} (\Omega)}$ provided that 3/2 <? q <? 3 and ${{F \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}}$ . Here L _q,r (Ω) denotes the well-known Lorentz space and q* =? 3q /(3 ? q) is the Sobolev exponent to q.     

Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds

In this paper, we consider the generalized solutions of the inequality $$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$ on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality ? Δu ≥ u ^q, q > 1, on spherically symmetric noncompact Riemannian manifolds.     

Sign-definiteness of q-eigenfunctions for a super-linear p-Laplacian eigenvalue problem

We consider the following q-eigenvalue problem for the p-Laplacian $$\left\{\begin{array}{ll}-{\rm div}\big( |\nabla u|^{p-2}\nabla u\big) = \lambda \|u\|_{L^{q}(\Omega)}^{p-q}|u|^{q-2}u \quad \quad\, {\rm in} \,\,\,\, \Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,\,{\rm on } \,\,\,\, \partial\Omega,\end{array}\right.$$ where \({\lambda\in\mathbb{R},}\) p > 1, Ω is a bounded and smooth domain of \({\mathbb{R}^{N},}\) N > 1, \({1\leq q < p^{\star}}\) , \({p^{\star}=\frac{Np}{N-p}}\) if p < N and \({p^{\star}=\infty}\) if \({p\geq N.}\) Let λ _q denote the first q-eigenvalue. We prove that in the super-linear case, \({p < q < p^{\star},}\) there exists \({\epsilon_{q}>0}\) such that if \({\lambda\in(\lambda_{q},\lambda _{q}+\epsilon_{q})}\) is a q-eigenvalue, then any corresponding q-eigenfunction does not change sign in Ω. As a consequence of this result we obtain, in the super-linear case, the isolatedness of λ _q for those Ω such that the Lane–Emden problem $$\left\{\begin{array}{ll}-{\rm div}\big(|\nabla u|^{p-2}\nabla u\big) = |u|^{q-2}u \qquad\quad\quad\quad \,\,{\rm in}\,\,\,\Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,{\rm on } \,\,\, \partial\Omega,\end{array}\right.$$ has exactly one positive solution.     

The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up?

The three-dimensional incompressible Euler equations with a passive scalar θ are considered in a smooth domain $\varOmega\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions $\boldsymbol{u}\cdot\hat{\boldsymbol{n}}|_{\partial\varOmega} = 0$ . It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B=?q×?θ, provided B has no null points initially: $\boldsymbol{\omega} = \operatorname{curl}\boldsymbol {u}$ is the vorticity and q=ω??θ is a potential vorticity. The presence of the passive scalar concentration θ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (Phys. Fluids 12:744–746, 2000) on the non-existence of Clebsch potentials in the neighbourhood of null points.     

Degenerate homogeneous parabolic equations associated with the infinity-Laplacian

We prove existence and uniqueness of viscosity solutions to the degenerate parabolic problem ${u_t = \Delta_{\infty}^{h} u}$ , where ${\Delta_{\infty}^{h}}$ is the h-homogeneous operator associated with the infinity-Laplacian, ${\Delta_{\infty}^{h} u = |Du|^{h-3} \langle D^{2}uDu, Du \rangle}$ , and h > 1. We also derive the asymptotic behaviour of u for the problem posed in the whole space, and for the Dirichlet problem posed in a bounded domain with zero boundary conditions.     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.     

The probabilistic solution of the dirichlet problem for\tfrac{1}{2}\Delta + \left\langle {a,\nabla } \right\rangle + b with singular coefficients

We investigate weak solutions ofLu=?p, whereL= \(\tfrac{1}{2}\Delta + \left\langle {a,\nabla } \right\rangle + b\) and |a|²,b andp belong to the Kato classK ^d (d≥3). We shall characterize the weak solutions by a probabilistic mean value property. This characterization includes a continuity principle. Our second regularity result states that for a weak solutionu, |?u|² belongs locally to K^d. Regarding the inhomogeneous Dirichlet problem forL, we shall prove the corresponding gauge theorem and an existence and uniqueness result.