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Исследуется вопрос об однозначной разрешимости первой смешанной задачи для вырождающегося параболического уравнения 2го порядка в случае, когда граничная и начальная функции принадлежат пространствам типа L2. Данная тематика берет начало с классических работ Ф. Рисса 1 и Литтлвуда и Пэли 2, посвященных граничным значениям аналитических функций. Дальнейшее развитие этой тематики для равномерно эллиптических уравнений получило в работах В. П. Михайлова, А. К. Гущина 3 9. Условие гладкости границы (Q C2) можно ослабить (см. 7). При наиболее слабых ограничениях на гладкость границы (и на коэффициенты уравнения) критерий существования граничного значения установлен в 7 9. При этом, как показано в работе 9, все направления принятия граничных значений для равномерно эллиптических уравнений оказываются равноправными, решение обладает свойством, аналогичным свойству непрерывности по совокупности переменных. В случае вырождения уравнения на границе области, когда направления не являются равноправными, ситуация более сложная. При этом постановка первой краевой задачи определяется типом вырождения. В случае, когда значения соответствующей квадратичной формы вырождающегося эллиптического уравнения на векторе нормали отличны от нуля (вырождение типа Трикоми), корректна задача Дирихле, и свойства такого вырождающегося уравнения весьма близки к свойствам равномерно эллиптического уравнения. В частности, в этой ситуации справедливы аналоги теорем Рисса 1 и Литтлвуда Пэли 2, 3. В случае вырождения типа Келдыша ситуация более сложная. Постановка первой краевой задачи и поведение решения вблизи границы определяются порядком вырождения уравнения, а в случае сильного вырождения коэффициентами при младших членах. Вопросам разрешимости первой краевой задачи для вырождающихся эллиптических и параболических уравнений посвящено большое число работ. Достаточно отметить работы Ф. Трикоми 10, М. В. Келдыша11, А. В. Бицадзе 12, С. А. Терсенова 13, И. М. Петрушко 14, 15, О. А. Олейник, Е. В. Радкевича 16, Г. Фикеры 17 и др. Из недавних работ можно отметить 18. В настоящей работе рассматривается случай сильного вырождения параболического уравнения 2го порядка, когда соответствующая квадратичная форма убывает как r(x) и постановка первой смешанной задачи определяется коэффициентом при первой производной по нормали., The paper investigates the question of the unique solvability of the first mixed problem for a degenerate secondorder parabolic equation in the case when the boundary and initial functions belong to spaces of type L2. This topic originates from the classical works of F. Riesz 1 and Littlewood and Paley 2 devoted to the boundary values of analytic functions. Under the weakest restrictions on the smoothness of the boundary (and on the coefficients of the equation), the criterion for the existence of a boundary value was established in 7 9. The boundary smoothness condition (Q C2) can be weakened (see 10). Moreover, as shown in 9, all directions of the adoption of boundary values for uniformly elliptic equations turn out to be equal the solution has the property similar to the property of continuity in the set of variables. In the case of degeneration of the equation at the boundary of the region, when the directions are not equal, the situation is more complicated. Moreover, the formulation of the first boundary value problem is determined by the type of degeneracy. In the case when the values of the corresponding quadratic form of the degenerate elliptic equation on the normal vector are nonzero (Tricomi type degeneracy), the Dirichlet problem is correct, and the properties of such degenerate equation are very close to the properties of a uniformly elliptic equation. In particular, in this situation, analogs of the Riesz theorems 1 and Littlewood Paley 2, 3 are valid. In the case of degeneracy of the Keldysh type, the situation is more complicated. The statement of the first boundaryvalue problem and the behavior of the solution near the boundary are determined by the order of degeneracy of the equation, and in the case of strong degeneracy, by the coefficients of the lower terms. A large number of papers have been devoted to the solvability of the first boundaryvalue problem for degenerate elliptic and parabolic equations. Note the works of F. Tricomi 11, M. V. Keldysh 12, A. V. Bitsadze 13, S. A. Tersenov 14, I. M. Petrushko 15, O. A. Oleinik and E. V. Radkevich 16, G. Fichera 17, etc. From recent works it is worth noting 18. In this paper, we consider the case of strong degeneracy of a secondorder parabolic equation when the corresponding quadratic form decreases as r(x) and the formulation of the first mixed problem is determined by the coefficient of the first normal derivative., №3(103) (2019)

Let (M, g) be a complete non-compact Riemannian manifold without boundary. In this paper, we give the gradient estimates on positive solutions to the following elliptic equation with singular nonlinearity: $$ \Delta u\left( x \right) + cu^{ - \alpha } \left( x \right) = 0 in M $$ , where α > 0, c are two real constants. When c < 0 and M is a bounded smooth domain in ℝn, the above equation is known as the thin film equation, which describes a steady state of the thin film (see Guo-Wei [Manuscripta Math., 120, 193–209 (2006)]). The results in this paper can be viewed as an supplement of that of J. Li [J. Funct. Anal., 100, 233–256 (1991)], where the nonlinearity is the positive power of u.

In the half-plane {−∞ < x < +∞} × {0 < y < +∞}, the Dirichlet problem is considered for differential-difference equations of the kind $$ {u}_{xx}+\sum_{k=1}^m{a}_k{u}_{xx}\left(x+{h}_{k,}y\right)+{u}_{yy}=0 $$ , where the amount m of nonlocal terms of the equation is arbitrary and no commensurability conditions are imposed on their coefficients a1, . . . , am and the parameters h1, . . . , hm determining the translations of the independent variable x. The only condition imposed on the coefficients and parameters of the studied equation is the nonpositivity of the real part of the symbol of the operator acting with respect to the variable x. Earlier, it was proved that the specified condition (i.e., the strong ellipticity condition for the corresponding differential-difference operator) guarantees the solvability of the considered problem in the sense of generalized functions (according to the Gel’fand–Shilov definition), a Poisson integral representation of a solution was constructed, and it was proved that the constructed solution is smooth outside the boundary line. In the present paper, the behavior of the specified solution as y → +∞ is investigated. We prove the asymptotic closeness between the investigated solution and the classical Dirichlet problem for the differential elliptic equation (with the same boundary function as in the original nonlocal problem) determined as follows: all parameters h1, . . . , hm of the original differential-difference elliptic equation are assigned to be equal to zero. As a corollary, we prove that the investigated solutions obey the classical Repnikov–Eidel’man stabilization condition: the solution stabilizes as y → +∞ if and only if the mean value of the boundary function over the interval (−R,+R) has a limit as R → +∞.

In this paper, we study the elliptic system of competitive type with nonhomogeneous terms $ \Delta u = u^pv^q+h_1(x) $, $ \Delta v = u^rv^s+h_2(x) $ in $ \Omega $ with two types of boundary conditions: (Ⅰ) $ u = v = +\infty $ and (SF) $ u = +\infty $, $ v = f $ on $ \partial\Omega $, where $ f > 0 $, $ (p-1)(s-1)-qr > 0 $, and $ \Omega \subset \mathbb{R^N} $ is a smooth bounded domain. The nonhomogeneous terms $ h_1(x) $ and $ h_2(x) $ may be unbounded near the boundary and may change sign in $ \Omega $. First, for a single semilinear elliptic equation with a singular weight and nonhomogeneous term, boundary asymptotic behaviour of large positive solutions is established. Using this asymptotic behaviour, we show existence of large positive solutions for this elliptic system with the boundary condition (SF), existence of maximal solution, boundary asymptotic behaviour and uniqueness of large positive solutions for this elliptic system with (Ⅰ).

A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For the second-order elliptic equation −div(α∇u) = f, this framework employs four different discretization variables, uh, ph, ŭh and $${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over p} _h}$$ , where uh and ph are for approximation of u and p = −α∇u inside each element, and ŭh and $${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over p} _h}$$ are for approximation of residual of u and p · n on the boundary of each element. The resulting 4-field discretization is proved to satisfy two types of inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, many existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.

In this paper, we shall investigate a semilinear elliptic boundary blow-up problem $$\Delta u=a(x)|u|^{p-1}u+h(x)$$ in $$\Omega $$ and $$u|_{\partial \Omega }=\infty $$, where $$\Omega $$ is a smooth bounded domain of $$\mathbb {R}^{N}$$. The weight a(x) and the nonhomogeneous term h(x) may be unbounded near the boundary. Furthermore, h(x) may change sign and a(x) may vanish in $$\Omega $$. The existence of a large solution for the problem under some assumptions on a(x) and h(x), and a consequent nonexistence result are established. We also prove the uniqueness of the solution.

This paper studies gradient estimates for positive solutions of the nonlinear elliptic equation $$\begin{aligned} \Delta _V(u^p)+\lambda u=0,\quad p\ge 1, \end{aligned}$$on a Riemannian manifold (M, g) with k-Bakry–Emery Ricci curvature bounded from below. We consider both the case where M is a compact manifold with or without boundary and the case where M is a complete manifold.

We consider the Dirichlet problem for an elliptic equation with a singularity. The singularity of the solution to the problem is caused by the presence of a re-entrant corner at the boundary of the domain. We define an R&nu, generalized solution for this problem. This allows for the construction of numerical methods for finding an approximate solution without loss of accuracy. In this paper, the existence and uniqueness of the R&nu, generalized solution in set W∘2,&alpha, 1(&Omega, &delta, ) is proven. The R&nu, generalized solution is the same for different parameters &nu

В данной работе, являющейся продолжением [1], устанавливаются необходимые и достаточные условия того, чтобы решение параболического уравнения 2-го порядка с боковой границей, принадлежащей классу $C^{1+\lambda}$, $\lambda >0$, вырождающегося на границе области, имело предел в среднем на боковой поверхности цилиндрической области и предел в среднем на ее нижнем основании, и исследуется вопрос об однозначной разрешимости первой смешанной задачи для такого уравнения в случае, когда граничная и начальная функции принадлежат пространствам типа $L_2$. Наиболее близкими к рассматриваемому кругу вопросов являются теоремы Ф. Рисса и Ж. Литтлвуда и Р. Пэли, в которых даются критерии предельных значений в $L_p$, $p > 1$, аналитических в единичном круге функций. Дальнейшее развитие этой тематики для равномерно эллиптических уравнений получило в работах В. П. Михайлова, А. К. Гущина [2–4]. Условие гладкости границы ($\partial Q \in C^2$) можно ослабить (см. [5]). При наиболее слабых ограничениях на гладкость границы (и на коэффициенты уравнения) критерии существования граничного значения установлены в работах [4, 6–8]. При этом, как показано в работе [7], все направления принятия граничных значений для равномерно эллиптических уравнений оказываются равноправными, решение обладает свойством, аналогичным свойству непрерывности по совокупности переменных. В случае вырождения уравнения на границе области, когда направления не являются равноправными, ситуация более сложная. При этом постановка первой краевой задачи определяется типом вырождения. В случае, когда значения соответствующей квадратичной формы вырождающегося эллиптического уравнения на векторе нормали отличны от нуля (вырождение типа Трикоми), корректна задача Дирихле, и свойства такого вырождающегося уравнения весьма близки к свойствам равномерно эллиптического уравнения. В частности, в этой ситуации справедливы аналоги теорем Рисса [9] и Литтлвуда — Пэли [10, 11]., This work, being a continuation of [1], establishes the necessary and sufficient conditions for the solution of the second-order parabolic equations with a lateral boundary from the class $C^{1+\lambda}$, $\lambda >0$, degenerating on the boundary of the domain, to have an average limit on the lateral surface of the cylindrical domain and the limit in the mean on its lower base. Also, we study the question of the unique solvability of the first mixed problem for such equations in the case when the boundary and initial functions belong to spaces of the типа $L_2$ type. The closest to the questions under consideration are the theorems of F. Riesz and J. Littlewood and R. Paley, in which criteria are given for the limit values in $L_p$, $p > 1$, of the functions analytic in the unit disk. Further development of this topic for uniformly elliptic equations was obtained in the papers by V. P. Mikhailov and A. K. Gushchin [2–4]. The boundary smoothness condition ($\partial Q \in C^2$) can be weakened (see [5]). Under the weakest restrictions on the smoothness of the boundary (and on the coefficients of the equation), the criteria for the existence of a boundary value were established in [4, 6–8]. In this case, as shown in [7], all directions of the acceptance of boundary values for uniformly elliptic equations turn out to be equal, while the solution has a property similar to the property of continuity with respect to the set of variables. In the case of degeneracy of the equation on the boundary of the domain when the directions are not equal, the situation is more complicated. In this case, the formulation of the first boundary value problem is determined by the type of degeneracy. In the case when the values of the corresponding quadratic form of the degenerate elliptic equation on the normal vector are different from zero (the Tricomi type degeneracy), the Dirichlet problem is correct, and the properties of such degenerate equations are very close to the properties of uniformly elliptic equations; in particular, in this situation analogues of the Riesz [9] and Littlewood–Paley theorems [10, 11] are valid., Журнал «Математические заметки СВФУ», Выпуск 2 (106) 2020

In this paper we study, in an open bounded set $\Omega\subset\mathbb R^N$ with Lipschitz boundary $\partial\Omega$, the Dirichlet problem for a nonlinear singular elliptic equation involving the $1$--Laplacian and a total variation term, that is, the inhomogeneous case of the equation appearing in the level set formulation of the inverse mean curvature flow. Our aim is twofold. On the one hand, we consider data belonging to the Marcinkiewicz space $L^{N,\infty}(\Omega)$, which leads to unbounded solutions. So, we have to begin introducing the suitable notion of unbounded solution to this problem. Moreover, examples of explicit solutions are shown. On the other hand, this equation allows us to deal with many related problems having a different gradient term. It is known that the total variation term induces a regularizing effect on existence, uniqueness and regularity. We focus on analyzing whether those features remain true when general gradient terms are taken. Roughly speaking, the bigger $g$, the better the properties of the solution.

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior, which is uniformly Lipschitz and nonlinear terms, are concentrated in a region, which neighbors the boundary of domain. We prove that this family of solutions converges to the solutions of a limit problem in H1an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.Copyright © 2015 John Wiley & Sons, Ltd.

This paper deals with the boundary-value problem for a nonlinear elliptic equation containing a small parameter multiplying the derivatives and degenerating into a finite equation as the small parameter tends to zero. The existence theorem for the solution with a boundary layer and its Lyapunov stability are proved.

Let (M, g) be a compact Riemannian manifold of dimension n $$(n\ge 2)$$ without boundary. In this paper, a fixed point result and a version of the Trudinger–Moser inequality are employed to establish sufficient conditions for the existence of solutions of quasilinear nonhomogeneous elliptic equation of the type $$\begin{aligned} - {{\mathrm{div}}}_g(a(x,\nabla _g u))+|u|^{n-2}u=f(x,u)+ \lambda h\quad \text {in }\quad M. \end{aligned}$$ We assume that there exists a function $$A : M \times T(M) \rightarrow \mathbb {R}$$ such that $$a(x,\xi ) = \nabla _{g} A(x,\xi )$$ satisfies the so-called Leray–Lions conditions. Here $$\lambda $$ is a positive parameter, the nonlinearity $$f: M \times \mathbb {R} \rightarrow \mathbb {R}$$ can be discontinuous and enjoy exponential critical growth and h belongs to the dual space of $$W^{1,n}(M)$$ with $$h\not \equiv 0$$ .

In this paper, we are interested in the existence and multiplicity results of solutions for the singular quasilinear elliptic problem with concave–convex nonlinearities (0.1) where is an unbounded exterior domain with smooth boundary ∂Ω, 1 0,1

The aim of this paper is study the equation −Δu = (log(u) + λu p )χ{u>0} in Ω with Dirichlet boundary condition, where 0 0 is sufficiently large, the corresponding energy functional to the perturbed equation −Δu + ge(u) = λ(u + ) p has nontrivial critical points ue in H 1 (Ω). Letting e → 0, then ue converges to a solution of the original problem, which is nontrivial and nonnegative. For 1

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 ⊂ ℝn, n ≥ 3, where 0 ≤ V (x) ∈ L1(Ω), 0 ≤ ϕ(x) ∈ L1(∂Ω) and ϕ(x) is continuous on the boundary ∂Ω. It is proved that there exists a constant C*(n) = (n − 2)2/4 such that if V0(x) = (c + λ1)|x|−2, then, for 0 ≤ c ≤ C*(n) and V(x) ≤ V0(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ϕ(x) ∈ L1(∂Ω); for c > C*(n) and V(x) ≥ V0(x) in Ω, this problem has no nonnegative solutions if ϕ(x) > 0.

This paper deals with the blow-up rate and uniqueness of large solutions of the elliptic equation \({\Delta u = b(x)f(u)+c(x)g(u)|\nabla u|^q}\) in \({\Omega \subset \mathbb{R}^N}\), where q > 0, f(u) and g(u) are regularly varying functions at infinity, and the weight functions \({b(x),\,c(x) \in C^\alpha(\Omega,\,\mathbb{R}^+)}\), 0 < α < 1, may be singular or degenerate on the boundary \({\partial\Omega}\). Combining the regular variation theoretic approach of Cirstea–Radulescu and the systematic approach of Bandle–Giarrusso, we are able to improve and generalize most of the previously available results in the literature.

We study the eigenvalue problem$\(-\sum_{i=1}^N\di\partial_{x_i}(|\di\partial_{x_i}u |^{p_i(x)-2}\di\partial_{x_i}u)$= λ|u|q(x)−2uin Ω,u= 0 on ∂Ω, where Ω is a bounded domain in ℝNwith smooth boundary ∂Ω, λ is a positive real number, andp1,⋅ ⋅ ⋅,pN,qare continuous functions satisfying the following conditions: 2 ≤pi(x)

LetΩ\Omegabe a bounded domain inRn,\mathbf {R}^n,n≥3,n \ge 3,with a boundary∂Ω∈C2.\partial \Omega \in C^2.We consider the following singularly perturbed nonlinear elliptic problem onΩ\Omega:\[ε2Δu−u+f(u)=0, u>0 on Ω,u=0 on ∂Ω,\varepsilon ^2 \Delta u - u + f(u) = 0, \ \ u > 0 \textrm { on }\Omega , \quad u = 0 \textrm { on } \partial \Omega ,\]where the nonlinearityffis of subcritical growth. Under rather strong conditions onf,f,it has been known that for smallε>0,\varepsilon > 0,there exists a mountain pass solutionuεu_\varepsilonof above problem which exhibits a spike layer near a maximum point of the distance functionddfrom∂Ω\partial \Omegaasε→0.\varepsilon \to 0.In this paper, we construct a solutionuεu_\varepsilonof above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions onff, which we believe to be almost optimal.

This paper is devoted to the existence and regularity of the homogenous Dirichlet boundary value problem for a singular nonlinear elliptic equation with natural growth in the gradient. By certain transformations, the problem can be transformed formally into either a Dirichlet problem or boundary blowup problems without gradient term, for which the corresponding existence results are also derived, which is a partial extension and supplement to the previous works. Copyright © 2008 John Wiley & Sons, Ltd.

This paper studies the asymptotic behavior near the boundary for large solutions of the semilinear equation Δu + au = b(x)f(u) in a smooth bounded domain Ω of ℝN with N ≥ 2, where a is a real parameter and b is a nonnegative smooth function on $$\overline \Omega $$ . We assume that f(u) behaves like u(ln u)α as u → ∞, for some α > 2. It turns out that this case is more difficult to handle than those where f(u) grows like u p (p > 1) or faster at infinity. Under suitable conditions on the weight function b(x), which may vanish on ∂Ω, we obtain the first order expansion of the large solutions near the boundary. We also obtain some uniqueness results.

Let N ≥ 4 N\geq 4 , let 2 ∗ = 2 N / ( N − 2 ) 2^{\ast }=2N/(N-2) and let Ω ⊂ R N \Omega \subset \mathbb {R}^{N} be a bounded domain with a smooth boundary ∂ Ω \partial \Omega . Our purpose in this paper is to consider the existence of solutions of the problem: \[ { − Δ u − μ u | x | 2 a m p ; = | u | 2 ∗ − 1 a m p ; a m p ; in Ω , u a m p ; > 0 a m p ; a m p ; in Ω , u a m p ; = 0 a m p ; a m p ; on ∂ Ω , \left \{ \begin {aligned} -\Delta u - \mu \frac {u}{\vert x\vert ^2} &= \vert u\vert ^{2^\ast -1} && \text {in $\Omega $}, \\ u & > 0 && \text {in $\Omega $},\\ u & = 0 && \text {on $\partial \Omega $}, \end {aligned} \right . \] where 0 > μ > ( N − 2 2 ) 2 . 0>\mu >(\frac {N-2}{2})^{2}.

A version of a famous and important result due to Alt-Caffarelli, relevant to the analysis of elliptic free boundary problems, states that there exists δ n > 0 such that if Ω ⊂ R n is an unbounded δ-Reifenberg flat domain, δ ⊂ (0, δ n ), and if ∂Ω satisfies an Ahlfors condition, then the following is true. Assume that there exist functions u (Green function with pole at infinity) and k (the Poisson kernel) such that Δu = 0 in Q, u > 0 in Q, u = 0 on ∂Ω and dω = kdH n-1 where ω is the harmonic measure at infinity. If furthermore sup X∈Ω |∇u(X)| ≤ 1 and k(Q) ≥ 1 for H n-1 a.e. point Q ∈ ∂Ω, then in suitable coordinates, Ω = {(x,x n ): x n > 0} and u(x, x n ) = x n . This result is crucial in recent work on the analysis of elliptic free boundary problems beyond the continuous threshold by Kenig and Toro. In this paper we consider the corresponding parabolic problems in the setting of time-varying domains Q = {(x o ,x,t) ∈ R x R n-1 x R: x 0 > ψ(x,t)} where ψ is a Lip(1, 1/2) function. Defining Ω 1 = Ω and Ω 2 = R n+1 \ Ω, we let ω i (X i , t i , ·), for i ∈ {1,2} and (X i , t i ) ∈ Ω i be the caloric measure defined with respect to Ω i . Assuming that ω i (X i , t i , ·) is absolutely continuous with respect to an appropriate surface measure σ for at least one i ∈ {1,2}, we study the implication of the condition log k i (X i , t i , ·) ∈ VMO(dσ) on the 'free boundary' ∂Ω. We show that this information on the Poisson kernel(s) can be explored in a delicate blow-up argument and that results on the regularity of ∂Ω can be deduced from classification theorems for global solutions to parabolic free boundary problems appearing in the limit. In fact, we prove a number of such classification theorems and, in particular, we prove weaker parabolic analogues of the result of Alt-Caffarelli.

In this paper, we study inhomogeneous Dirichlet problems for elliptic equations in divergence form. Optimal regularity requirements on the coefficients and domains for theW1,p (1>p>∞)W^{1,p}\ (1>p>\infty )estimates are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO semi-norms. The domain is supposed to have Lipschitz boundary with small Lipschitz constant. These conditions for theW1,pW^{1,p}theory do not just weaken the requirements on the coefficients; they also lead to a more general geometric condition on the domain.

In this paper we study the existence of nontrivial solutions for the following system of two coupled semilinear Poisson equations: formula math. where Ω is a bounded domain in R 2 with smooth boundary ∂Ω, and the functions f and g have the maximal growth which allow us to treat problem (S) variationally in the Sobolev space H 1 0 (Ω). We consider the case with nonlinearities in the critical growth range suggested by the so-called Trudinger-Moser inequality.

Let Ω be a bounded, smooth domain in R 2n , n > 2. The well-known Moser-Trudinger inequality ensures the nonlinear functional Jp(u) is bounded from below if and only if ρ ≤ ρ 2n := 2 2n n!(n - 1)!w 2n , where J ρ (u) =1/2 ) f (-△) n/2 u| 2 - ρ log f e u dx in x:= H n (Ω) ∩ {u, (-△) j u ∈HΩ (Ω, j = 1,...[ n-1 / 2 ]), and ω 2n is the area of the unit sphere S 2n-1 in R 2n . In this paper, we prove the inf u ∈ x Jp(u) is always attained for p < ρ 2n . The existence of minimizers of Jp at the critical value p = ρ 2n is a delicate problem. The proof depends on the blowup analysis for a sequence of bubbling solutions. Here we develop a local version of the method of moving planes to exclude the boundary bubbling. The existence of minimizers for Jp at the critical value p = ρ 2n is in contrast to the case of two dimensions.

In this paper, we are concerned with the zero Dirichlet boundary value problem associated to the quasilinear elliptic equation [Formula: see text] where [Formula: see text] is an open and bounded set in [Formula: see text] ([Formula: see text]), [Formula: see text] is a continuously differentiable real function in [Formula: see text], [Formula: see text] is an elliptic, bounded and symmetric matrix, [Formula: see text] is non-negative and may be singular at zero and [Formula: see text]. We give sufficient conditions on [Formula: see text], [Formula: see text] and [Formula: see text] in order to have a comparison principle and, as a consequence, uniqueness of positive solutions being continuous up to the boundary.

Consider a function u, harmonic in a ring-shaped domain and taking two constant (distinct) values on the two connected components of the boundary. If we know in advance that one of the components is a sphere, and that u satisfies some overdetermined condition on the other one, can we conclude that u is radial? This paper answers this question for certain overdetermined conditions on the gradient of u, generalizing some previous results. Conditions depending on the principal curvatures of the boundary are also investigated. Existence and uniqueness of a radial solution to the overdetermined problem are discussed. Some extensions to ellipsoidal domains, as well as to quasilinear elliptic equations, are carried out.

For a given nonlinear partial differential equation defined on a bounded domain with irregular boundary, the available analytical tools are very limited in relation to the study of positive solutions. In this paper wer first use weak convergence methods to show that for an elliptic equation of a certain type, classical positive solutions on nearby smooth domains approach a generalised positive solution on the given domain. The idea is then applied to sublinear elliptic problems to obtain existence and uniqueness results.

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.

In this paper we consider the group K4(3)(F) of a field, in case F is the function field of a smooth geometrically irreducible curve over a number field. We do this using the complexes constructed in [4], together with an auxiliary complex. On the image in K4(3) (F) of those complexes, we derive a formula for the Beilinson regulator, and compute an approximation of the boundary map at the closed points of the curve in the localization sequence. We give a way of finding examples of elliptic curves E with elements in K4(3)(E), and in some cases use computer calculations to check numerically the relation between the regulator and the L-function, as conjectured by Beilinson.

The quasi-hydrodynamical model for bipolar semiconductor devices in thermal equilibrium admits in general solutions for which the non-negative electron- or hole-density is not strict positive. In this paper sufficient conditions depending on the device's data which lead to or prevent vacuum are presented. The transformation of the model equations to a single semilinear elliptic mixed boundary value problem for the electrostatic potential V reduces the vacuum-non-vacuum discussion to an investigation of the range of V. Monotonicity arguments and the employment of local sub- and supersolutions allow to estimate the size as well as the distance from the Dirichlet boundary of the respective vacuum sets. An one-dimensional model is analysed in some detail.

In this paper, we consider the existence of minimizers of the Hardy-Sobolev type variational problem. Recently, Ghoussoub and Robert (9, 10) proved that the Hardy- Sobolev best constant admits its minimizers provided the bounded smooth domain has the negative mean curvature at the origin on the boundary. We generalize their results by using the idea of Brezis and Nirenberg, and as a consequence, we shall prove the existence of pos- itive solutions to the elliptic equation involving two different kinds of Hardy-Sobolev critical exponents.

In this paper, we obtain a version of the sliding plane method of Gidas, Ni and Nirenberg which applies to domains with no smoothness condition on the boundary. The method obtains results on the symmetry of positive solutions of boundary value problems for nonlinear elliptic equations. We also show how our techniques apply to some problems on half spaces.

This paper analyzes the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form $\nabla \cdot D^\eps \nabla u^\eps=0$ or $(-\Delta + q^\eps) u^\eps=0$ with prescribed Neumann conditions. The theory is well-known when the constitutive parameters in the elliptic equation assume the values of different and smooth functions in the background and inside the inclusions. We generalize the results to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions of the trace of the solution at the domain's boundary up to an order $\eps^{2d}$, where $d$ is dimension and $\eps$ is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most $\eps^{d+1}$ rather than the expected $\eps^d$. We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables., Comment: 42 pages

We study the following two phase elliptic singular perturbation problem: Δ u e = β e ( u e ) + f e , in Ω ⊂ R N , where e > 0 , β e ( s ) = 1 e β ( s e ) , with β a Lipschitz function satisfying β > 0 in ( 0 , 1 ) , β ≡ 0 outside ( 0 , 1 ) and ∫ β ( s ) d s = M . The functions u e and f e are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit ( e → 0 ) and we show that limit functions are solutions to the two phase free boundary problem: Δ u = f χ { u ≢ 0 } in Ω ∖ ∂ { u > 0 } , | ∇ u + | 2 − | ∇ u − | 2 = 2 M on Ω ∩ ∂ { u > 0 } , where f = lim f e , in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case f e ≡ 0 . The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary.

We solve variationally certain equations of stellar dynamics of the form $-\sum_i\partial_{ii} u(x) =\frac{|u|^{p-2}u(x)}{{\rm dist} (x,{\mathcal A} )^s}$ in a domain $\Omega$ of $\rn$, where ${\mathcal A} $ is a proper linear subspace of $\rn$. Existence problems are related to the question of attainability of the best constant in the following recent inequality of Badiale-Tarantello [1]: $$0, Comment: 27 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.pims.math.ca/~nassif/

In this paper, we deal with a class of semilinear elliptic equation in a bounded domain $\Omega\subset\mathbb{R}^N$, $N\geq 3$, with $C\sp{1,1}$ boundary. Using a new fixed point result of the Krasnoselskii's type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical., Comment: 4 pages

A novel, very effective Liapunov functional was used in previous papers to derive decay and asymptotic stability estimates (with respect to time) in a variety of thermal and thermo-mechanical contexts. The purpose of this note is to show that the versatility of this functional extends to certain non-linear elliptic boundary value problems in a right cylinder, the axial co-ordinate in this context replacing the time variable in the previous one. A steady-state temperature problem is considered with Dirichlet boundary conditions, the condition on the boundary being independent of the axial co-ordinate. The functional is used to obtain an estimate of the error committed in approximating the temperature field by the two-dimensional field induced by the boundary condition on the lateral surface. Copyright © 2002 John Wiley & Sons, Ltd.

SynopsisIn this paper the boundary behaviour of the solution of Dirichlet's problem for the equation Δu = f in domains with C2,α-boundary is studied. Without using Schauder's a priori estimates, we prove the following theorem, which is indispensable in Schauder's method to solve Dirichlet's problem for a general linear elliptic equation.Theorem: Let (f, g)∈ C0,α (Ḡ)×C2,α (∂G) and u ∈ C0.0(Ḡ) ∩C2,α(G) be a solution of the problem Δu = f, u|∂G = g. Then u ∈ C2,α(Ḡ).

SynopsisThe object of this paper is to demonstrate, that with the open mapplng theorem of S.Banach one can prove very easily the following estimatefor all u ∈ C2,α and 0 ≤ t ≤ 1, if one knows, that for all bounded G ⊂ Rn, with boundary ∂G ∈ C2,α and for all (f, g) ∈ C0,α × C2,α (∂G) Dirichle's problem Δu = f, u|∂G = g has a solution u ∈ C2,α. This estimate can be used to solve Dirichle's problem for a general linear elliptic equation by Schauder's method.