On the Estimate of the Absolute Value of Eigenfunctions to the Steklov–Zaremba Problem for the Laplace Operator

Abstract

The Steklov–Zaremba problem for the Laplace operator in a bounded domain with a strictly Lipschitz boundary is considered. A homogeneous Dirichlet condition is specified on the closed part of the boundary of the domain, and the Steklov boundary condition with a spectral parameter is assumed to be satisfied on the complement to the closed part. This problem is a natural generalization of the classical Steklov problem. With respect to a closed set on the boundary of the domain, where the homogeneous Dirichlet boundary condition is specified, its Wiener capacity is assumed to be positive. It follows from this condition on the capacity that it is natural to consider the problem in the Sobolev space of functions that are square-integrable together with all generalized (weak) first-order derivatives. The aim of the paper is to find an estimate for the maximum modulus of normalized eigenfunctions to the problem under consideration. The proofs of the main results make substantial use of the iterative technique of Jurgen Moser.

1. Introduction

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n,n\ge 2$ be a bounded strictly Lipschitz domain. The classical Steklov problem has the following form:

$$-\mathrm{\Delta }u=0\mathrm{in}\mathrm{\Omega },{\displaystyle \frac{\partial u}{\partial \nu }}{|}_{\partial \mathrm{\Omega }}=\lambda u,$$

where $\lambda$ is the spectral parameter. The first systematic study of this problem was carried out in [1] (see also [2]).

A domain $\mathrm{\Omega }$ is called strictly Lipschitz if for each point $x_0\in \partial \mathrm{\Omega }$ there exists an open cube Q centered at $x_0$ whose faces are parallel to the coordinate axes, the edge length does not depend on $x_0$, and in some Cartesian coordinate system with the origin at $x_0$ the set $Q\cap \partial \mathrm{\Omega }$ is the graph of a Lipschitz function $x_n=g(x_1,\dots ,x_{n-1})$ with a Lipschitz constant L independent of $x_0$.

This paper is devoted to a generalization of the Steklov problem, which we call the Steklov–Zaremba problem.

Before we formulate the Steklov–Zaremba problem, we define the Sobolev space of functions $W_2^1(\mathrm{\Omega },F)$. Here $F\subset \partial \mathrm{\Omega }$ is a closed set, $W_2^1(\mathrm{\Omega },F)$ is the completion of infinitely differentiable in the closure of $\mathrm{\Omega }$ functions that are equal to zero in a neighborhood of F, with respect to the norm of the space $W_2^1\left(\mathrm{\Omega }\right)$ of the following form:

$${\Vert v\Vert }_{W_2^1(\mathrm{\Omega },F)}={\left(\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{2}dx+\underset{\mathrm{\Omega }}{\int }{|\nabla v|}^{2}dx\right)}^{1/2}$$

with the positive constant $C<\infty$. A priori, the Friedrichs inequality is required for functions $v\in W_2^1(\mathrm{\Omega },F)$:

$$\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{2}dx\le C\underset{\mathrm{\Omega }}{\int }{|\nabla v|}^{2}dx.$$

(1)

We formulate a necessary and sufficient condition on the set $F\subset \partial \mathrm{\Omega }$ that guarantees the inequality (1) is satisfied. For this, we need the concept of capacity.

By ${\mathcal{Q}}_d$ we denote an open cube with edge length d and faces parallel to the coordinate axes, assuming that the strictly Lipschitz domain $\mathrm{\Omega }$ has diameter d and $\mathrm{\Omega }\subset {\mathcal{Q}}_d$. Let us introduce the concept of capacity $C_2(K,{\mathcal{Q}}_{2d})$ of the compact set $K\subset {\overline{\mathcal{Q}}}_d$ with respect to the cube ${\mathcal{Q}}_{2d}$ by the following equality:

$$C_2(K,{\mathcal{Q}}_{2d})=inf\left\{\underset{{\mathcal{Q}}_{2d}}{\int }{|\nabla \psi |}^{2}dx:\psi \in C_0^{\infty }\left({\mathcal{Q}}_{2d}\right),\psi \ge 1\mathrm{on}K\right\}.$$

From Maz’ia’s results (see [3] [§14.1.2] and comments to the results of Chapter 14 of the monograph [3]), it follows that for functions $v\in W_2^1(\mathrm{\Omega },F)$ the inequality (1) holds then and only then when the following is met:

$$C_2(F,{\mathcal{Q}}_{2d})>0.$$

(2)

In particular, if the set F is the closure of a connected set containing interior points, then the condition (2) is satisfied automatically. Examples of closed sets F of the Cantor set type are constructed in [4] ($n=2$) and [5] ($n=3$).

Our aim is to obtain the estimate of the maximum of the modulus of eigenfunctions to the Steklov–Zaremba problem, calculated as follows:

$${-\mathrm{\Delta }u=0,u|}_F=0,{\displaystyle \frac{\partial u}{\partial \nu }}{|}_G=\lambda u,$$

(3)

normalized by

$$\underset{\mathrm{\Omega }}{\int }{u}^{2}dx=1.$$

(4)

Here $G=\partial \mathrm{\Omega }\setminus F$, and $\nu =({\nu }_1,\dots ,{\nu }_n)$ is the unit outward normal to the boundary $\partial \mathrm{\Omega }$ of the domain $\mathrm{\Omega }$.

The solution to the problem (3) is understood to be a function $u\in W_2^1(\mathrm{\Omega },F)$ for which the integral identity, written as follows:

$$\underset{\mathrm{\Omega }}{\int }\nabla u·\nabla \psi dx=\lambda \underset{G}{\int }u\psi ds,$$

(5)

holds true on all functions $\psi \in W_2^1(\mathrm{\Omega },F)$. In this case the values of $\lambda$ are called eigenvalues.

Since the operator of the problem is self-adjoint in $W_2^1(\mathrm{\Omega },F)$ and positive, under the above assumptions on the domain $\mathrm{\Omega }$, the problem (3) has a complete orthonormal system of eigenfunctions, which are defined in the weak sense. All eigenfunctions correspond to positive eigenvalues. Note that problem (3) was considered in the paper [6], where an estimate of the counting function was obtained under special restrictions on the domain and set F. An estimate of the maximum modulus of the eigenfunction of problem (3) is obtained in this paper.

The proof of the estimate of the maximum modulus of the eigenfunctions of the Steklov–Zaremba problem is based on the technique of estimating the maximum modulus of the eigenfunctions of the Dirichlet problem for an operator of the following form:

$$Lu=\sum _{i,j=1}^n{\displaystyle \frac{\partial }{\partial x_i}}\left(a_{ij}\left(x\right){\displaystyle \frac{\partial u}{\partial x_j}}\right),$$

(6)

where the matrix $a_{ij}$ satisfies the ellipticity condition

$${\alpha \left|\xi \right|}^2\le \sum _{i,j=1}^n{a}_{ij}{\xi }_i{\xi }_j\le {\displaystyle \frac{1}{\alpha }}{\left|\xi \right|}^2.$$

There are estimates of the eigenfunctions of the Dirichlet problem in a strictly Lipschitz bounded domain, written as follows:

$$Lu=\lambda u\mathrm{in}\mathrm{\Omega },u=0\mathrm{on}\partial \mathrm{\Omega },$$

and numerous studies have been devoted to this (see [7,8,9,10,11]). In particular, the work of V.A. Ilyin and I.A. Shishmarev [10] showed that if the domain and coefficients $a_{ij}\left(x\right)$ are sufficiently smooth, then for the classical eigenfunction $u_k\left(x\right)$ corresponding to the eigenvalue ${\lambda }_k$, the estimate, written as follows:

$$\underset{x\in \overline{\mathrm{\Omega }}}{max}\left|u_k\left(x\right)\right|\le C(n,\mathrm{\Omega },\alpha ){\lambda }_k^{{\displaystyle \frac{n}{4}}}$$

is valid, where $\alpha$ is the ellipticity constant of the operator L.

If the coefficients $a_{ij}\left(x\right)$ of the operator (6) are measurable, then in [11] (see also [12]) it was proved that the estimate for the generalized eigenfunction, written as follows:

$$\underset{x\in \mathrm{\Omega }}{\mathrm{ess}\sup }\left|u\left(x\right)\right|\le C(n,\mathrm{\Omega },\alpha ){\lambda }^{{\displaystyle \frac{n}{4}}}$$

is satisfied.

Later, in the work [13] for an operator of the form (6) with sufficiently smooth coefficients and in a domain with a sufficiently smooth boundary, a more precise estimate was established

$$\underset{x\in \mathrm{\Omega }}{\mathrm{ess}\sup }\left|u\left(x\right)\right|\le C(n,\mathrm{\Omega },\alpha ){\lambda }^{{\displaystyle \frac{n-1}{4}}}.$$

If we consider an ordinary second-order differential equation ($n=1$), then the analogous result (boundedness of the modulus of the eigenfunction) was obtained in the work [14].

We are interested in estimating the maximum of the modulus of the eigenfunctions to the problem (3) for sufficiently large values of $\lambda$. In particular, we assume that $\lambda \ge 1$. The main result of this paper is the following statement.

Theorem 1.

If condition (2) is satisfied, then under assumption (4) for the eigenfunctions of problem (3) the estimate, written as follows:

$$\underset{x\in \mathrm{\Omega }}{\mathrm{ess}\sup }\left|u\left(x\right)\right|\le C(n,\mathrm{\Omega },F){\lambda }^{{\displaystyle \frac{n}{2}}}$$

(7)

holds true with the constant C depending only on n, the domain Ω and the compact F.

The proof of Theorem 1 for $n>2$ is a consequence of the following assertion by (4).

Theorem 2.

If the conditions of Theorem 1 are satisfied, then the inequality, written as follows:

$$\underset{x\in \mathrm{\Omega }}{\mathrm{ess}\sup }\left|u\left(x\right)\right|\le C(\alpha ,n,\mathrm{\Omega },F){\lambda }^{n/4}{\left(\underset{\mathrm{\Omega }}{\int }{u}^{2}dx\right)}^{1/2}$$

(8)

holds for the solution u of the problem (3).

The proof of both theorems is based on the well-known iterative technique of J. Moser (see [15]).

We have already mentioned that Friedrichs inequality (1) holds for functions in the space $W_2^1(\mathrm{\Omega },F)$. Because of this, a norm can be defined in the space $W_2^1(\mathrm{\Omega },F)$ that does not contain the square of the function itself. Next, we use Sobolev embedding theorems for strictly Lipschitz domains, meaning a norm containing only the gradient of the function. We assume that condition (2) holds, which implies the validity of Friedrichs inequality (1).

2. Estimate of the Modulus of Eigenfunctions: Case $\mathit{n}>\mathbf{2}$

In this section we prove Theorem 2, which directly implies the result of Theorem 1.

Below, for functions $\psi \in W_2^1(\mathrm{\Omega },F)$, we use the Sobolev embedding theorem, written as follows (see, for example, [3] [§14.1]):

$${\left(\underset{\mathrm{\Omega }}{\int }{\left|\psi \right|}^{2k}dx\right)}^{{\displaystyle \frac{1}{k}}}\le C(n,\mathrm{\Omega },F)\underset{{\mathrm{\Omega }}^i}{\int }{|\nabla \psi |}^{2}dx,\psi \in W_2^1(\mathrm{\Omega },F),k={\displaystyle \frac{n}{n-2}}.$$

(9)

In addition, we need a theorem on the embedding of traces of functions from $W_1^1\left(\mathrm{\Omega }\right)$ (see [16] [Ch. II, §2]) of the following form:

$$\underset{\partial \mathrm{\Omega }}{\int }\left|\psi \right|ds\le C(n,\mathrm{\Omega })\underset{\mathrm{\Omega }}{\int }\left(\left|\psi \right|+|\nabla \psi |\right)dx,\psi \in W_1^1\left(\mathrm{\Omega }\right).$$

(10)

Proof. 

Assume the following:

$$u_{+}\left(x\right)=\left\{\begin{array}{cc}u\left(x\right),\hfill & \mathrm{if}u\left(x\right)>0,\hfill \\ 0,\hfill & \mathrm{if}u\left(x\right)\le 0\hfill \end{array}\right.$$

and

$$u_{-}\left(x\right)=\left\{\begin{array}{cc}-u\left(x\right),\hfill & \mathrm{if}u\left(x\right)<0,\hfill \\ 0,\hfill & \mathrm{if}u\left(x\right)\ge 0.\hfill \end{array}\right.$$

(11)

For $\beta \ge 1$, we set $T_j\left(\zeta \right)={\zeta }^{\beta }$ for $\zeta \in [0,j]$, $T_j\left(\zeta \right)=j^{\beta }+\beta j^{\beta -1}(\zeta -j)$ for $\zeta \ge j$, $D_j\left(\zeta \right)=\underset{0}{\overset{\zeta }{\int }}{\left|T_j^{\prime }\left(\xi \right)\right|}^{2}d\xi$.

Choosing in (5) a test–function $\psi \left(x\right)=D_j\left(u_{+}\left(x\right)\right)$, which is admissible, we have the following:

$$\underset{\mathrm{\Omega }}{\int }{|\nabla u_{+}|}^2{D}_j^{\prime }\left(u_{+}\left(x\right)\right)dx=\lambda \underset{G}{\int }{u}_{+}{D}_j\left(u_{+}\left(x\right)\right)ds=\lambda \underset{\partial \mathrm{\Omega }}{\int }{u}_{+}{D}_j\left(u_{+}\left(x\right)\right)ds.$$

(12)

Due to (10), we derive the following:

$$\lambda \underset{\partial \mathrm{\Omega }}{\int }{u}_{+}{D}_j\left(u_{+}\left(x\right)\right)ds\le C(n,\mathrm{\Omega })\lambda \left(\underset{\mathrm{\Omega }}{\int }{u}_{+}{D}_j\left(u_{+}\left(x\right)\right)dx+\underset{\mathrm{\Omega }}{\int }|\nabla (|u_{+}{D}_j\left(u_{+}\left(x\right)\right|dx\right)\le$$

$$\le C(n,\mathrm{\Omega })\lambda \left(\underset{\mathrm{\Omega }}{\int }{u}_{+}{D}_j\left(u_{+}\left(x\right)\right)dx+\underset{\mathrm{\Omega }}{\int }|\nabla u_{+}|D_j(u_{+}\left(x\right)dx+\underset{\mathrm{\Omega }}{\int }|\nabla u_{+}|u_{+}{D}_j^{\prime }\left(u_{+}\left(x\right)\right)dx\right).$$

Next, by the Cauchy inequality with $\delta >0$, we obtain the following:

$$\lambda \underset{\mathrm{\Omega }}{\int }|\nabla u_{+}|D_j(u_{+}\left(x\right)dx\le \delta \lambda \underset{\mathrm{\Omega }}{\int }|\nabla u_{+}{|}^2{D}_j^{\prime }\left(u_{+}\left(x\right)\right)dx+{\displaystyle \frac{\lambda }{4\delta }}\underset{\mathrm{\Omega }}{\int }{\left(D_j^{\prime }\left(u_{+}\left(x\right)\right)\right)}^{-1}{D}_j^2(u_{+}\left(x\right)dx,$$

$$\lambda \underset{\mathrm{\Omega }}{\int }|\nabla u_{+}|u_{+}{D}_j^{\prime }\left(u_{+}\left(x\right)\right)dx\le \delta \lambda \underset{\mathrm{\Omega }}{\int }{|\nabla u_{+}|}^2{D}_j^{\prime }\left(u_{+}\left(x\right)\right)dx+{\displaystyle \frac{\lambda }{4\delta }}\underset{\mathrm{\Omega }}{\int }{u}_{+}^2{D}_j^{\prime }\left(u_{+}\left(x\right)\right)dx.$$

Taking into account the previous relations in (12), after the appropriate choice of $\delta$, we obtain the following:

$$\begin{array}{cc}\hfill \underset{\mathrm{\Omega }}{\int }{|\nabla u_{+}|}^2{D}_j^{\prime }\left(u_{+}\left(x\right)\right)dx\le C(n,\mathrm{\Omega })(& \lambda \underset{\mathrm{\Omega }}{\int }{u}_{+}{D}_j\left(u_{+}\left(x\right)\right)dx+{\lambda }^2\underset{\mathrm{\Omega }}{\int }{u}_{+}^2{D}_j^{\prime }\left(u_{+}\left(x\right)\right)dx+\hfill \\ \hfill & +{\lambda }^2\underset{\mathrm{\Omega }}{\int }{\left(D_j^{\prime }\left(u_{+}\left(x\right)\right)\right)}^{-1}{D}_j^2(u_{+}\left(x\right)dx).\hfill \end{array}$$

The following is assumed to be valid:

$$\begin{array}{cc}\hfill \underset{\mathrm{\Omega }}{\int }{|\nabla T_j\left(u_{+}\left(x\right)\right)|}^{2}dx\le C(n,\mathrm{\Omega })(& \lambda \underset{\mathrm{\Omega }}{\int }{u}_{+}{D}_j\left(u_{+}\left(x\right)\right)dx+{\lambda }^2\underset{\mathrm{\Omega }}{\int }{u}_{+}^2{D}_j^{\prime }\left(u_{+}\left(x\right)\right)dx+\hfill \\ \hfill & +{\lambda }^2\underset{\mathrm{\Omega }}{\int }{\left(D_j^{\prime }\left(u_{+}\left(x\right)\right)\right)}^{-1}{D}_j^2(u_{+}\left(x\right)dx).\hfill \end{array}$$

From here, using Sobolev’s inequality (9) we find the following:

$$\begin{array}{cc}\hfill {\left(\underset{\mathrm{\Omega }}{\int }\left(T_j{\left(u_{+}\left(x\right)\right)}^{2k}\right)dx\right)}^{{\displaystyle \frac{1}{k}}}\le C(n,\mathrm{\Omega },F)(& \lambda \underset{\mathrm{\Omega }}{\int }{u}_{+}{D}_j\left(u_{+}\left(x\right)\right)dx+{\lambda }^2\underset{\mathrm{\Omega }}{\int }{u}_{+}^2{D}_j^{\prime }(u_{+}\left(x\right))dx+\hfill \\ \hfill & +{\lambda }^2\underset{\mathrm{\Omega }}{\int }{\left(D_j^{\prime }\left(u_{+}\left(x\right)\right)\right)}^{-1}{D}_j^2(u_{+}\left(x\right)dx).\hfill \end{array}$$

(13)

Passing to the limit as $j\to +\infty$ and using the fact that $\lambda \ge 1$, we obtain the following:

$$\begin{array}{cc}\hfill & {\left(\underset{\mathrm{\Omega }}{\int }{\left(u_{+}\left(x\right)\right)}^{2k\beta })dx\right)}^{{\displaystyle \frac{1}{k}}}\le \hfill \\ \hfill & \le C(n,\mathrm{\Omega },F)\left({\displaystyle \frac{{\beta }^2}{2\beta -1}}\lambda +{\beta }^2{\lambda }^2+{\displaystyle \frac{{\beta }^2}{{(2\beta -1)}^2}}{\lambda }^2\right)\underset{\mathrm{\Omega }}{\int }{\left(u_{+}\left(x\right)\right)}^{2\beta }dx\le \hfill \\ \hfill & \le C(n,\mathrm{\Omega },F){\beta }^2{\lambda }^2\underset{\mathrm{\Omega }}{\int }{\left(u_{+}\left(x\right)\right)}^{2\beta }dx.\hfill \end{array}$$

(14)

If we carry out similar reasoning to the previous ones with the function $u_{+}$ replaced by $u_{-}$ (see (11)), then we have the following:

$$\begin{array}{c}\hfill {\left(\underset{\mathrm{\Omega }}{\int }{\left(u_{-}\left(x\right)\right)}^{2k\beta })dx\right)}^{{\displaystyle \frac{1}{k}}}\le C(n,\mathrm{\Omega },F){\beta }^2{\lambda }^2\underset{\mathrm{\Omega }}{\int }{\left(u_{-}\left(x\right)\right)}^{2\beta }dx.\end{array}$$

(15)

By comparing the inequalities (14) and (15), we find the following:

$$\begin{array}{c}\hfill {\left(\underset{\mathrm{\Omega }}{\int }{\left|u\left(x\right)\right|}^{2k\beta })dx\right)}^{{\displaystyle \frac{1}{k}}}\le C(n,\mathrm{\Omega },F){\beta }^2{\lambda }^2\underset{\mathrm{\Omega }}{\int }{\left|u\left(x\right)\right|}^{2\beta }dx\end{array}$$

or

$$\begin{array}{c}\hfill {\left(\underset{\mathrm{\Omega }}{\int }{\left|u\left(x\right)\right|}^{2k\beta })dx\right)}^{{\displaystyle \frac{1}{2\beta k}}}\le {\left(C(n,\mathrm{\Omega },F)\right)}^{{\displaystyle \frac{1}{2\beta }}}{\beta }^{{\displaystyle \frac{1}{\beta }}}{\lambda }^{{\displaystyle \frac{1}{\beta }}}{\left(\underset{\mathrm{\Omega }}{\int }{\left|u\left(x\right)\right|}^{2\beta }dx\right)}^{{\displaystyle \frac{1}{2\beta }}}.\end{array}$$

(16)

Let us iterate this estimate. For $i=0,1,\dots$ we set $\beta =k^i$. By means of the estimate (16) considering the following:

$${\mathrm{{\rm Y}}}_i={\left(\underset{\mathrm{\Omega }}{\int }{\left|u\right|}^{2k^i}dx\right)}^{{\displaystyle \frac{1}{2k^i}}},$$

we obtain the recurrence relation

$${\mathrm{{\rm Y}}}_{i+1}\le {\left(C(n,\mathrm{\Omega },F)\right)}^{{\displaystyle \frac{1}{2k^i}}}{\left(k^i\right)}^{{\displaystyle \frac{1}{k^i}}}{\lambda }^{{\displaystyle \frac{1}{k^i}}}{\mathrm{{\rm Y}}}_i.$$

Hence, by induction (see [15]) we obtain the following:

$${\mathrm{{\rm Y}}}_{i+1}\le \prod _{m=0}^i{\left(C(n,\mathrm{\Omega },F)\right)}^{{\displaystyle \frac{1}{2k^m}}}{\left(k^m\right)}^{{\displaystyle \frac{1}{k^m}}}{\lambda }^{{\displaystyle \frac{1}{k^m}}}{\mathrm{{\rm Y}}}_0.$$

Or, since $k=n/(n-2)$, we have the following:

$$\underset{x\in \mathrm{\Omega }}{\mathrm{ess}\sup }\left|u\left(x\right)\right|=\underset{i\to \infty }{lim}{\mathrm{{\rm Y}}}_i\le C(n,\mathrm{\Omega },F){\lambda }^{n/2}{\mathrm{{\rm Y}}}_0,$$

which yields the desired inequality (8). The Theorem 1 is proved. □

3. Estimate of the Absolute Value of Eigenfunctions: Case $\mathit{n}=\mathbf{2}$

In this section we prove Theorem 1 in the planar case.

In the case $n=2$, for functions $\psi \in W_2^1(\mathrm{\Omega },F)$, the Sobolev embedding theorem is read as follows:

$${\left(\underset{\mathrm{\Omega }}{\int }{\left|\psi \right|}^{q}dx\right)}^{2/q}\le C(n,\mathrm{\Omega },F)\underset{\mathrm{\Omega }}{\int }{|\nabla \psi |}^{2}dx,\psi \in W_2^1(\mathrm{\Omega },F),q\ge 1.$$

(17)

Note that the solution u to problem (3) in the case $n=2$ is bounded. This can be easily verified if in (17) we consider, for example, $q=4$, then, following the proof from the previous section, we obtain the inequality, written as follows:

$$\underset{x\in \mathrm{\Omega }}{\mathrm{ess}\sup }\left|u\left(x\right)\right|\le C(n,\mathrm{\Omega },F){\lambda }^2{\left(\underset{\mathrm{\Omega }}{\int }{u}^{2}dx\right)}^{1/2}.$$

Since the solution is bounded in $\mathrm{\Omega }$ and hence, continuous (see Theorem 8.22 [17] [§8.9]), there exists a point $x_0$ in $\mathrm{\Omega }$ such that the following exists:

$$\underset{\mathrm{\Omega }}{\mathrm{ess}\sup }\left|u\left(x\right)\right|\ge |u\left(x_0\right)|-\delta$$

(18)

for some positive $\delta$. Let us make a change of variables (homothy)

$$y=(x-x_0)\lambda ,v\left(y\right)=u(x_0+{\displaystyle \frac{y}{\lambda }}),$$

where $y\in \tilde{\mathrm{\Omega }}$, and $\tilde{\mathrm{\Omega }}$ is the image of the domain $\mathrm{\Omega }$ under such a transformation. Problem (3) can be rewritten as follows:

$$\mathrm{\Delta }v=0\mathrm{in}\tilde{\mathrm{\Omega }}{,v|}_{\tilde{F}}=0,{\displaystyle \frac{\partial v}{\partial \nu }}{|}_{\tilde{G}}=v,$$

(19)

where ${\displaystyle \frac{\partial v}{\partial \nu }}$ is the normal derivative, and $\tilde{F}$ and $\tilde{G}$ are the images of F and G under this transformation. The solution is understood in the generalized (weak) sense, i.e., $v\in W_2^1(\tilde{\mathrm{\Omega }},\tilde{F})$ is a solution if the integral identity, written as follows:

$$\underset{\tilde{\mathrm{\Omega }}}{\int }\nabla v·\nabla \psi dy=\underset{\tilde{G}}{\int }v\psi ds$$

(20)

is valid for any function $\psi \in W_2^1(\tilde{\mathrm{\Omega }},\tilde{F})$. Note that the solution v remains bounded. Denote by $B_R$ the open ball of radius R centered at the origin. Let us show the following:

$$\underset{B_1\cap \tilde{\mathrm{\Omega }}}{\mathrm{ess}\sup }\left|v\left(y\right)\right|\le C(\mathrm{\Omega },F){\left(\underset{\tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{2}dy\right)}^{{\displaystyle \frac{1}{2}}}.$$

(21)

We choose $\psi ={v\left|v\right|}^{\beta -1}{\eta }^2$ as a test function in the identity (20), where $\beta \ge 1$ and $\eta \in C_0^{\infty }\left(B_2\right)$, $0<\eta <1$. This test function is admissible, because the function v is bounded. Finally, we obtain the following:

$$\begin{array}{c}\hfill \beta \underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{|\nabla v|}^2{\left|v\right|}^{\beta -1}{\eta }^{2}dy=-2\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }\nabla v·{\nabla \eta v\left|v\right|}^{\beta -1}\eta dy+\underset{B_4\cap \tilde{G}}{\int }{\left|v\right|}^{\beta +1}{\eta }^{2}ds.\end{array}$$

(22)

From here, using standard reasoning, we derive the relation as follows:

$$\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{|\nabla v|}^2{\left|v\right|}^{\beta -1}{\eta }^{2}dy\le C\left(\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{\beta +1}{|\nabla \eta |}^{2}dy+\underset{B_4\cap \tilde{G}}{\int }{\left|v\right|}^{\beta +1}{\eta }^{2}ds\right).$$

Applying the inequality (10) and the Cauchy inequality, we obtain the following:

$$\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{|\nabla v|}^2{\left|v\right|}^{\beta -1}{\eta }^{2}dy\le C(\mathrm{\Omega },F)\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{\beta +1}\left({|\nabla \eta |}^2+|\nabla \eta |+{\eta }^2\right)dy.$$

Rewriting the left-hand side of the inequality, we obtain the following:

$$\begin{array}{c}\hfill {\displaystyle \frac{4}{{(\beta +1)}^2}}\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{|\nabla (\left|v\right|}^{{\displaystyle \frac{\beta +1}{2}}}{\left)\right|}^2{\eta }^{2}dy\le C(\mathrm{\Omega },F)\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{\beta +1}\left({|\nabla \eta |}^2+|\nabla \eta |+{\eta }^2\right)dy.\end{array}$$

(23)

Now, using the obvious inequality, written as follows:

$${|\nabla \left(\psi \zeta \right)|}^2\le {2\left(\right|\nabla \psi |}^2{\zeta }^2+{\psi }^2{|\nabla \zeta |}^2),$$

from (23) we obtain

$$\begin{array}{c}\hfill \underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{|\nabla (\left|v\right|}^{{\displaystyle \frac{\beta +1}{2}}}{\eta \left)\right|}^{2}dy\le C(\mathrm{\Omega },F){(\beta +1)}^2\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{\beta +1}\left({|\nabla \eta |}^2+|\nabla \eta |+{\eta }^2\right)dy.\end{array}$$

(24)

We use the Sobolev embedding theorem in the domain $B_4\cap \tilde{\mathrm{\Omega }}$, according to which we have the following:

$${\left(\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{\left(\right|v|}^{{\displaystyle \frac{\beta +1}{2}}}{\eta )}^{q}dy\right)}^{{\displaystyle \frac{2}{q}}}\le C(\mathrm{\Omega },F)\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{|\nabla (\left|v\right|}^{{\displaystyle \frac{\beta +1}{2}}}\eta {\left)\right|}^{2}dy,q\ge 1,$$

from (24) we have

$$\begin{array}{c}\hfill {\left(\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{\left(\right|v|}^{{\displaystyle \frac{\beta +1}{2}}}{\eta )}^{q}dy\right)}^{{\displaystyle \frac{2}{q}}}\le C(\mathrm{\Omega },F){(\beta +1)}^2\underset{B_4\cap \tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{\beta +1}\left({|\nabla \eta |}^2+|\nabla \eta |+{\eta }^2\right)dy.\end{array}$$

(25)

Now we use the cutoff function $\eta$ in (25). For $j=0,1,\dots$ we have $R_j=1+2^{-j}$ and let $\eta \in C_0^{\infty }\left(B_{R_j}\right)$, $0<\eta <1$, $\eta =1$ in $B_{R_{j+1}}$ and $|\nabla \eta |\le const·2^j$. Choosing $q=4$ in the Sobolev embedding theorem and putting $\beta +1=2^{j+1}={\chi }_j$, from (25) we derive the following:

$$\begin{array}{c}\hfill {\left(\underset{B_{R_{j+1}}\cap \tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{{\chi }_{j+1}}dy\right)}^{{\displaystyle \frac{1}{{\chi }_{j+1}}}}\le C(\mathrm{\Omega },F){\left({\chi }_j\right)}^{{\displaystyle \frac{2}{{\chi }_j}}}{2}^{{\displaystyle \frac{2j}{{\chi }_j}}}{\left(\underset{B_{R_j}\cap \tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{{\chi }_j}dy\right)}^{{\displaystyle \frac{1}{{\chi }_j}}}.\end{array}$$

(26)

Hence, by induction (see [15]) we have the following:

$${\left(\underset{B_{R_{j+1}}\cap \tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{{\chi }_{j+1}}dy\right)}^{{\displaystyle \frac{1}{{\chi }_{j+1}}}}\le \prod _{m=0}^{j}C(\mathrm{\Omega },F){\left({\chi }_m\right)}^{{\displaystyle \frac{2}{{\chi }_m}}}{2}^{{\displaystyle \frac{2m}{{\chi }_m}}}{\left(\underset{B_2\cap \tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{2}dy\right)}^{{\displaystyle \frac{1}{2}}}.$$

Passing to the limit at $j\to +\infty$, we arrive at the estimate (21). From this estimate, due to the choice of the point $x_0$ (see (18)), the following is assumed:

$$\underset{\tilde{\mathrm{\Omega }}}{\mathrm{ess}\sup }\left|v\left(y\right)\right|-\delta \le C(\mathrm{\Omega },F){\left(\underset{\tilde{\mathrm{\Omega }}}{\int }{\left|v\right|}^{2}dy\right)}^{{\displaystyle \frac{1}{2}}}.$$

(27)

Passing here to the limit in $\delta \to 0$ and returning to the original variables, keeping in mind (4) and the following:

$$\underset{\tilde{\mathrm{\Omega }}}{\int }{\left|v\left(y\right)\right|}^{2}dy=\lambda \underset{\mathrm{\Omega }}{\int }{\left|u\left(x\right)\right|}^{2}dx=\lambda .$$

(28)

we obtain the following

$$\underset{x\in \mathrm{\Omega }}{\mathrm{ess}\sup }\left|u\left(x\right)\right|\le C(\mathrm{\Omega },F)\lambda ,$$

(29)

which proves Theorem 1 for $n=2$.

4. Conclusions

In this paper, we consider the Steklov–Zaremba problem for the Laplace operator in a strictly Lipschitz bounded domain. On the closed part F of the domain boundary, the homogeneous Dirichlet condition is assumed to be satisfied, and on its complement G, the Steklov spectral condition is set. With respect to the closed subset F, its Wiener capacity is assumed to be greater than zero.

In this paper, we prove an estimate for the maximum of the modulus of the normalized eigenfunctions to the Steklov-Zaremba problem.

Note that Moser’s iterative technique was used to prove the main results.

The case $n=1$ is trivial, since the solutions are written explicitly. Therefore, the maximum of the modulus of the eigenfunctions is simply bounded.

We note some related results on this topic in the works [18,19,20,21].

Unlike the Steklov problem, the Steklov–Zaremba problem we are considering has been virtually unstudied and is poorly covered in the literature. There is only one known paper (see [6,22]) that considers a similar problem, but in a smooth domain. In this paper, the counting function was estimated, but the estimation of the eigenfunctions was not considered.

Note that this is the first time the Steklov–Zaremba problem has been considered in a strictly Lipschitz domain. Moreover, eigenfunctions for the Laplace operator with Zaremba conditions on the boundary exist (this follows from general operator theory) and will be infinitely differentiable inside the domain. We have not investigated the smoothness of the eigenfunctions, concentration, and localization in the vicinity of the boundary. Friedrichs inequality plays an important role, and it is formulated in terms of Wiener capacity. We use V.G. Maz’ya’s necessary and sufficient condition for this inequality to hold in a strictly Lipschitz domain.

In the presence of the Friedrichs inequality, the operator considered in the Sobolev space $W_2^1(\mathrm{\Omega },F)$ we introduced is coercive, and the set of orthogonal eigenfunctions is complete in the chosen Sobolev space, as follows from the operator’s positive definiteness and self-adjointness. The bilinear form corresponding to this problem is coercive.

We were not interested in the applied aspects of the Steklov-Zaremba problem, but such problems arise in the study of surface waves in vessels containing liquids covered by a mesh or a “leaky lid” (see [23]).

In this paper, we use the well-known classical iterative technique of Moser, which has played a significant role in studying the qualitative properties of solutions to second-order linear uniformly elliptic equations in divergence form. However, this technique is not applicable to higher-order equations.

An estimate for the eigenfunctions of the Zaremba problem for uniformly elliptic operators in divergence form was obtained in [24]. It is of the same order as the estimate obtained in this paper. This paper is the first result in this direction, and we cannot guarantee the accuracy of the estimate obtained. The question of future generalizations of these results is complex and has not yet been considered. The first step toward this should be to study the eigenfunctions of the Steklov-Zaremba problem for more general linear uniformly elliptic equations in divergence form with measurable coefficients.