Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities

1 Introduction

This is a brief description of results of [1–3] with comments and improvements, which were presented at the International conference on differential equations dedicated to the 110th anniversary of Ya. B. Lopatynsky. In these articles we have investigated the behaviour of solutions of quasi-linear elliptic problems in a neighborhood of boundary singularities in bounded and unbounded domains. We have found exponents of the solution’s decreasing rate of the type |u(x)| ≤ O(|x|^α), near the boundary singularities.

Let G ⊂ ℝⁿ be an unbounded domain (see Fig. 1) with boundary ∂G that is a smooth surface everywhere except at the origin 𝓞 and near 𝓞 it is a conical surface, n ≥ 2. We assume that G=G0R∪GR, where G0R=G0d∪GdR and G0d is a rotational cone with the vertex at 𝓞 and the aperture ω₀ ∈ (0,π), d ≪ 1, G_R = {x = (r, ω) ∈ ℝⁿ|r ∈ (R, ∞), ω ∈ Ω ⊂ S^{n − 1}, n ≥ 2}, R ≫ 1, S^{n − 1} is the unit sphere.

Fig. 1

an unbounded cone-like domain.

We introduce the following notations for a domain G which has a boundary conical point:

1. Ω : = G ∩ S^{n − 1}, where S^{n − 1} denotes the unit sphere in ℝⁿ;

2. ∂Ω: the boundary of Ω;

3. Gab := G ∩{(r, ω) : 0 ≤ a < r < b, ω ∈ Ω}: a layer in ℝⁿ;

4. Γab := ∂G ∩{(r, ω) : 0 ≤ a < r < b, ω ∈ ∂Ω}: the lateral surface of Gab ;

5. Gd:=G∖G0d,Γd:=∂G∖Γ0d.

We use standard function spaces: Ck(G¯) with the norm |u|k,G¯;Lp(G) with the norm ∥u∥_p,G, p ≥ 1; the Sobolev space W^k,p(G) with the norm ∥u∥_W^k,p(G) for integer k ≥ 0; the weighted Sobolev-Kondratiev space Vp,αk(G) for integer k ≥ 0, 1 < p < ∞ and α ∈ ℝ with the finite norm ∥u∥Vp,αk(G)=(∫G∑|β|≤krα+p(|β|−k)|Dβu|pdx)1p and the space Vp,αk,−1p(Γ) of functions φ, given on ∂G, with the norm ∥g∥Vp,αk,−1p(∂G)=inf∥G∥Vp,αk(G), where the infimum is taken over all functions g such that 𝓖|_∂G = g in the sense of traces.

2 Oblique derivative problem

In [1] we have investigated the behaviour of strong solutions to the oblique derivative problem for the general second-order quasi-linear elliptic equation in a neighbourhood of a conical boundary point of an n-dimensional bounded domain, n ≥ 2. In the case of the linear equation we refer to [4, 5].

Local maximum principle for strong solutions to the elliptic quasi-linear oblique derivative problem in convex rotational cones has been obtained by Lieberman in [6]. He [7] and Trudinger [8] have obtained local gradient bound estimate and local Hölder gradient estimate of strong solutions in any sub-domain with a C² boundary portion of the domain.

The results obtained in [1] are a generalization and improvement of results of [9] on the case of the oblique boundary condition. We consider the oblique derivative problem for the elliptic second-order linear equation:

where n→ denotes the unite exterior normal vector to ∂G0R∖O,(r,ω) are spherical coordinates in ℝⁿ with pole 𝓞; repeated indices are understood as summation from 1 to n.

3 Quasi-linear nonlocal Robin problem

In [3] we have investigated the behaviour of weak solutions for the nonlocal Robin problem with quasi-linear elliptic divergence second-order equations in a plain domain in a neighbourhood of the boundary corner point 𝓞. In the case of the linear equation we refer to [10, 11].

Here, we consider a different eigenvalue problem (see (EVP2)) and derive a new Friedrichs-Wirtinger type inequality adapted to the quasi-linear elliptic problem considered in [3]. It allows us to improve the main result of [3] (see Theorem 3.7): the exponent of weak solution behavior (see inequality (20)) in a neighborhood of an angular point 𝓞 in the case 𝓑 < 0 is better than that one obtained in [3].

We consider the type of nonlocal problems, where the support of nonlocal terms intersects the boundary. Namely, the situation in which a part Γ of domain boundary ∂G is mapped by transformation γ on γ(Γ) and γ(Γ)¯∩∂G≠∅.

Let us consider the domain G0R ⊂ ℝ². Moreover, let Γ₊ and Γ₋ be the part of boundary ∂ G0R for which x₂ > 0 and x₂ < 0 respectively. We assume, that ∂G0R=Γ¯+∪Γ¯− is a smooth curve everywhere except at the origin 𝓞 ∈ ∂ G0R and near the point 𝓞 curves Γ_± are lateral sides of an angle with the measure ω₀ ∈ [0, π) and the vertex at 𝓞. Let Σ₀ = G0R ∩{x₂ = 0}, where O∈Σ0¯. Furthermore, let γ be a diffeomorphism mapping of Γ₊ onto Σ₀. Additionally, we suppose that there exists d > 0 such that in the neighbourhood of Γ0+d the mapping γ is the rotation about the origin 𝓞 and −ω02 is the angle of rotation. It means γ(Γ0+d¯)=Σ0d¯=G0d¯∩Σ0.

We have considered a quasi-linear elliptic equation with the nonlocal boundary condition connecting the values of the unknown function u on the boudary part Γ₊ with its values of u on Σ₀ :

here q ≥ 0, m > 1, μ ≥ 0, a₀ ≥ 0, β_± > 0, b ≥ 0 are given numbers and n→ denotes the unit outward with respect to G0R normal to ∂G0R∖O.

Recall that we are dealing with the nonlocal problem and the boundary ∂ G0R is non smooth, so the formulation of problem (QL₂) does not make sense in general. To make our formulation precise, we give the definition of the weak solution of (QL₂).