ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Pages 88–93 (September 29, 2003) S 1079-6762(03)00115-X

ON ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF THE DIRICHLET PROBLEM IN HALF-SPACE

FOR LINEAR AND QUASI-LINEAR ELLIPTIC EQUATIONS

VASILY DENISOV AND ANDREY MURAVNIK

(Communicated by Michael E. Taylor)

Abstract. We study the Dirichlet problem in half-space for the equation Δu + g(u)|∇u|^{2} = 0, where g is continuous or has a power singularity (in the latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as y → ∞, where y denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function.

Introduction

The phenomenon called stabilization is well known for parabolic equations both in linear (see e.g. [1] and references therein) and non-linear (see e.g. [2] and references therein) cases; it means the existence of a finite limit of the solution as t → ∞. However, there are well-posed non-isotropic elliptic boundary-value problems in unbounded domains (see e.g. [3]) for which we can talk about stabilization in the following sense: the solution has a finite limit as a selected spatial variable tends to infinity.

This paper is devoted to the Dirichlet problem in half-space for elliptic equations. We present necessary and sufficient conditions for the stabilization of its solution; here the spatial variable, orthogonal to the hyperplane of boundary-value data, plays the role of time. In Section 1, the linear case is presented; Sections 2 and 3 are devoted to quasi-linear equations with the so-called Burgers-Kardar-Parisi-Zhang non-linearity type (see e.g. [4], [5]). Equations with such non-linearities arise, for example, in modeling of directed polymers and interface growth. They also present an independent theoretical interest because they contain second powers of the first derivatives (see e.g. [6] and references therein).

Note that we deal with the stabilization problem in cylindrical domains with an unbounded base (in particular, here the base of the cylinder is the whole E^{N}). As in the parabolic case, this problem is principally different (this refers both to the results and to the methods of research) from the stabilization problem in cylindrical domains with a bounded base. The latter problem has been investigated

\begin{matrix}c \\ ◯ \end{matrix}2003 American Mathematical Society

Received by the editors March 6, 2002.

2000 Mathematics Subject Classification. Primary 35J25; Secondary 35B40, 35J60.

Key words and phrases. Asymptotic behaviour of solutions, BKPZ-type non-linearities.

The second author was supported by INTAS, grant 00-136 and RFBR, grant 02-01-00312.

for a rather broad class of non-linear elliptic equations by means of methods of dynamical systems (see e.g. [7] and references therein).

1. Linear equations

In the half-space {y ≥ 0} we consider the Dirichlet problem

(1)  p(x)\frac{∂^{2}u}{∂y^{2}} + ∑_{i,j=1}^{N} \frac{∂}{∂x_{i}} (a_{ij}(x, y)\frac{∂u}{∂x_{j}} ) = 0, (x, y) ∈ \{y > 0\},

(2)  u(x, 0) = φ(x), x ∈ E^{N},

where the coefficients of equation (1) are measurable and bounded and the uniform ellipticity condition is satisfied, i.e., there exists a positive constant λ such that for any x, y ∈ {y > 0},

(3)  λ^{−1} ≤ p(x) ≤ λ, λ^{−1}|ξ|^{2} ≤ ∑_{i,j=1}^{N} a_{ij}(x, y)ξ_{i}ξ_{j} ≤ λ|ξ|^{2};

the boundary-value function φ(x) is supposed to be continuous and bounded. We say that u(x, y) is a solution of (1) if u is bounded on {y > 0}, belongs to W_{2}^{1} in any strictly interior subdomain of {y ≥ 0}, and satisfies the integral identity

∫_{0}^{+∞} ∫_{E^{N}} [ ∑_{i,j=1}^{N} a_{ij}(x, y)\frac{∂u}{∂x_{j}} \frac{∂v}{∂x_{i}} + p(x)\frac{∂u}{∂y} \frac{∂v}{∂y} ]dxdy = 0

for any v(x, y) ∈ C_{0}^{∞}{y > 0}.

It is known (see e.g. [3]) that, under the above assumptions, a solution u(x, y) of (1), (2) exists and is unique; it is bounded and continuous on {y > 0} and coincides at y = 0 with a continuous boundary-value function φ(x) on E^{N}. Moreover, it follows from [8] and [9] that u(x, y) satisfies Hölder’s condition with a constant α > 0 depending merely on λ and N.

We discuss the question of finding necessary and sufficient conditions for the existence of

(4)  lim_{y→+∞} u(x, y) = l

for any x ∈ E^{N} (or uniformly with respect to x ∈ E^{N}).

First of all we introduce a few definitions related to the means of multi-variable functions.

Let g(x) be bounded on E^{N}, let K be a compact subset of E^{N}, and let R be a large positive parameter. We consider the following family of functions:

{g(Rx)} = {g(Rx_{1}, Rx_{2}, …, Rx_{N})}, R > 0.

Suppose g(x) has a weak limit at infinity in L_{loc}^{2}, i.e.,

(5)  g(Rx) ⇀ \overline{g}(x),

where the symbol ⇀ denotes weak convergence in L^{2}(K). It follows from the boundedness of g(x) that (5) is equivalent to the following two assertions:

(6)  lim_{R→∞} ∫_{E^{N}} g(Rx)ψ(x) dx = ∫_{E^{N}} \overline{g}(x)ψ(x) dx for any ψ ∈ C_{0}^{∞}(E^{N}),

(7)  lim_{R→∞} ∫_{K} g(Rx) dx = ∫_{K} \overline{g}(x) dx.

We say that g(x) has a mean value, if it has a weak limit (5) at infinity in L_{loc}^{2} and that limit is a constant. Relation (7) yields that the above definition of the mean value is equivalent to the following one:

(8)  \overline{g} = lim_{R→∞} \frac{1}{|K_{R}|} ∫_{K_{R}} g(y) dy,

where K_{R} =^{def} {y ∈ E^{N} |y = Rx, x ∈ K}, |K_{R}| denotes the Lebesgue measure of K_{R}.

We say that g(x) has a uniform mean value \overline{g} on E^{N} if for any ψ ∈ C_{0}^{∞}(E^{N}) there exists

(9)  lim_{R→∞} ∫_{E^{N}} g(β + Rx)ψ(x) dx = \overline{g} ∫_{E^{N}} ψ(x) dx,

and this limit is uniform with respect to β ∈ E^{N}. The latter definition is equivalent to the following one:

(10)  lim_{R→∞} \frac{1}{|K_{R}|} ∫_{K_{R}} g(β + y) dy = \overline{g}

uniformly with respect to β ∈ E^{N}.

Let f(x, y) =^{def} f(x_{1}, …, x_{N}; y) be bounded on the half-space {y ≥ 0}. We say that it satisfies condition A if there exists a constant \overline{f} such that

(11)  lim_{R→∞} ∫_{|x|≤1} ∫_{0}^{1}[f(Rx, Ry) − \overline{f}]^{2} dy dx = 0.

Note that an A-type condition was introduced in [10] and [11]. It was developed further in [12].

Now we are ready to formulate our main results concerning linear equations.

Theorem 1. Suppose p(x) has a mean value and the elliptic operator in (1) is the Laplacian. Then limit (4) exists at any point x∈E^{N} if and only if

(12)  lim_{R→∞} \frac{∫_{|x|≤R} p(x)φ(x) dx}{∫_{|x|≤R} p(x) dx} = l.

Theorem 2. Assume p(x) has a mean value and the coefficients a_{ij}(x, y) satisfy condition A with a positive definite matrix \{\overline{a}_{ij}\}_{i,j=1}^{n}. Then limit (4) exists at any point x∈E^{N} if and only if

(13)  lim_{R→∞} \frac{∫_{(Bx,x)≤R^{2}} φ(x)p(x) dx}{∫_{(Bx,x)≤R^{2}} p(x) dx} = l,

where B denotes the inverse matrix of \{\overline{a}_{ij}\}_{i,j=1}^{n}.

Theorem 3. Limit (4) exists uniformly with respect to x∈E^{N} if and only if

(14)  lim_{R→∞} \frac{∫_{|y|≤R} p(x + y)φ(x + y) dy}{∫_{|y|≤R} p(x + y) dy} = l

uniformly with respect to x∈E^{N}.

Remark 1. In the case of Cauchy problem for parabolic equations, assertions similar to Theorems 1–3 were proved in [10]–[12].

2. Quasi-linear equations with regular coefficients at non-linearities

Hereinafter the point x = (x_{1}, …, x_{N}, x_{N+1}) is denoted by (x′, x_{N+1}), and the half-space E^{N} × (0, +∞) is denoted by E_{+}^{N+1}.

In this section we consider the following problem:

(15)  Δu + g(u)|∇u|^{2} = 0, x ∈ E_{+}^{N+1};
(16)  u(x′, 0) = φ(x′), x′ ∈ E^{N};

where g is continuous in (−∞, +∞), and φ is continuous and bounded in E^{N}. Following e.g. [13], we introduce the function

(17)  f(s) =^{def} ∫_{0}^{s} e^{∫_{0}^{τ} g(σ)dσ}dτ,

and prove the following assertions.

Theorem 4. There exists a unique classical bounded solution of problem (15), (16).

Theorem 5. Let x′ ∈ E^{N}, l ∈ (−∞, +∞), and let u(x) be the classical bounded solution of problem (15), (16). Then

lim_{x_{N+1}→+∞} u(x) = l if and only if lim_{R→+∞} \frac{NΓ(\frac{N}{2} )}{2π^{\frac{N}{2}} R^{N}} ∫_{|y|≤R} f[φ(y)]dy = f(l).

Theorem 6. Let l ∈ (−∞, +∞), and let u(x) be the classical bounded solution of problem (15), (16). Then

u(x) ⟶^{x_{N+1}→+∞} l uniformly with respect to x′ ∈ E^{N}

if and only if

\frac{NΓ(\frac{N}{2} )}{2π^{\frac{N}{2}} R^{N}} ∫_{|y|≤R}f[φ(x′ + y)]dy ⟶^{R→+∞}f(l) uniformly with respect to x′ ∈ E^{N}.

Remark 2. The results of this section are valid also for the equation

\frac{∂^{2}u}{∂y^{2}} +∑_{i,j=1}^{N} \frac{∂}{∂x_{i}} (a_{ij}(x, y)\frac{∂u}{∂x_{j}} ) + g(u)[ ∑_{i,j=1}^{N} a_{ij}(x, y)(\frac{∂u}{∂x_{j}} )^{2}+ (\frac{∂u}{∂y})^{2}] = 0,

where, besides (3) and condition A, we have a_{ij} ∈ C^{1,q}(\overline{E_{+}^{N}}) with a positive q (i, j = 1, …, N).

3. The case of singular coefficients at non-linearities

In this section, we study the equation

(18)  Δu + αu^{β}|∇u|^{2} = 0,

where β ∈ [−1, 0) and α > −1 for β = −1.

We will also assume that φ(x′) ≥ 0 (apart from its continuity and boundedness), and consider positive solutions of (18), (16).

For β ∈ (−1, 0) we use (17) again, but for β = −1 the function f(s) is defined as s^{α+1}. Then the following assertions are valid.

Theorem 7. If φ is different from the identical zero, then there exists a unique classical positive bounded solution of problem (18), (16).

Theorem 8. Let β = −1 and let u(x) be the classical positive bounded solution of problem (18), (16), x′ ∈ E^{N}, l ≥ 0. Then

lim_{x_{N+1}→+∞} u(x) = l if and only if lim_{R→+∞} \frac{NΓ(\frac{N}{2} )}{2π^{\frac{N}{2}} R^{N}} ∫_{|y|≤R} φ^{α+1}(y)dy = l^{α+1}.

Theorem 9. Let β ∈ (−1, 0) and let u(x) be the classical positive bounded solution of problem (18), (16), x′ ∈ E^{N}, l ≥ 0. Then

lim_{x_{N+1}→+∞} u(x) = l if and only if lim_{R→+∞} \frac{NΓ(\frac{N}{2} )}{2π^{\frac{N}{2}} R^{N}} ∫_{|y|≤R} f̃[φ(y)]dy = f(l),

where

(19)  f̃(s) = ∫_{0}^{s} e^{\frac{α}{1−β} τ^{1−β}}dτ.

Theorem 10. Let β = −1 and let u(x) be the classical positive bounded solution of problem (18), (16), l ≥ 0. Then

u(x) ⟶^{x_{N+1}→+∞} l uniformly with respect to x′ ∈ E^{N}

if and only if

\frac{NΓ(\frac{N}{2} )}{2π^{\frac{N}{2}} R^{N}} ∫_{|y|≤R}φ^{α+1}(x′ + y)dy ⟶^{R→+∞}l^{α+1} uniformly with respect to x′ ∈ E^{N}.

Theorem 11. Let β ∈ (−1, 0) and let u(x) be the classical positive bounded solution of problem (18), (16), l ≥ 0. Then

u(x) ⟶^{x_{N+1}→+∞} l uniformly with respect to x′ ∈ E^{N}

if and only if

\frac{NΓ(\frac{N}{2} )}{2π^{\frac{N}{2}} R^{N}} ∫_{|y|≤R} f̃[φ(x′ + y)]dy ⟶^{R→+∞} f̃(l) uniformly with respect to x′ ∈ E^{N}.

Remark 3. In the case of Cauchy problem for parabolic equations, assertions similar to Theorems 5–11 were proved in [2].

Acknowledgments

The authors are very grateful to L. A. Peletier and S. I. Pohožaev for fruitful discussions. The authors also thank V. A. Il’in, E. I. Moiseev and A. L. Skubachevskii for their attention and concern.

References

1. V. N. Denisov and V. D. Repnikov, The stabilization of a solution of a Cauchy problem for parabolic equations, Differ. Equ. 20, No. 1 (1984), 16–33. MR 86c:35013

2. V. N. Denisov and A. B. Muravnik, On stabilization of the solution of the Cauchy problem for quasilinear parabolic equations, Differ. Equ. 38, No. 3 (2002), 369–374.

3. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 1977. MR 57:13109

4. M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889–892.

5. E. Medina, T. Hwa, M. Kardar, and Y.-C. Zhang, Burgers equation with correlated noise: Renormalization group analysis and applications to directed polymers and interface growth, Phys. Rev. A39 (1989), 3053–3075. MR 89m:82027 6. S. I. Pohožaev, Equations of the type Δu = f(x, u, Du), Math. USSR Sb. 41, No. 2 (1982),

269–280. MR 82i:35066

7. A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Differ. Equations 110, No. 2 (1994), 322–355. MR 95f:35086

8. E. De Giorgi, Sulla differenziabilita e l’analitica delle estremali degli integrali multipli regolaru, Mat. Acad. Sci. Torino, Ser. 3, No. 1 (1957), 25–43. MR 20:172

9. N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR Izv. 16 (1981), 151–164. MR 83c:35059

10. A. K. Guscin and V. P. Mihailov, Stabilization of solutions of Cauchy’s problem for a parabolic equation, Differ. Equ. 7, No. 2 (1971), 232–242. MR 43:7758

11. S. Kamin, On stabilization of the solutions of the Cauchy problem for parabolic equations, Proc. Roy. Soc. Edinburgh 76A (1976), 43–53. MR 55:13079

12. V. V. Žikov, On the stabilization of solutions of parabolic equations, Math. USSR Sb. 33, No. 4 (1977), 519–537. MR 57:13190

13. A. V. Bitsadze, Some classes of partial differential equations, Nauka, 1981 (in Russian). MR 83e:35003

Moscow State University, Faculty of Computational Mathematics and Cybernetics, Leninskie gory, Moscow 119899, Russia

E-mail address: V.Denisov@g23.relcom.ru

Department of Differential Equations, Moscow State Aviation Institute, Volokolamskoe shosse 4, Moscow, A-80, GSP-3, 125993, Russia

E-mail address: abm@mailru.com