The purpose of the present paper is to derive Hadamard’s variational formulae for simple eigenvalue of the Laplacian and is to derive several new inequalities concerning the first eigenvalue concerning the mixed boundary condition.

In the previous work [9] we studied Hadamard’s variational formula for general domain deformation and extended the results [3] on two or three-space dimensions under normal perturbations of the domain. There, we develop an abstract theory of perturbation of self-adjoint operators, refining the argument in [6].

To be precise, let $(X,|\left|\right)$ and $(V,\parallel \parallel )$ be a pair of Hilbert spaces over $\mathbb{R}$ with compact embedding $V\hookrightarrow X$. Henceforth, $C>0$ denotes a generic constant. Let $A_t:V\times V\to \mathbb{R}$ and $B_t:X\times X\to \mathbb{R}$ for $t\in I=(-{\epsilon }_0,{\epsilon }_0)$, ${\epsilon }_0>0$, be symmetric bilinear forms satisfying and for some $\delta >0$. We take the abstract eigenvalue problem which ensures a sequence of eigenvalues denoted by

The associated normalized eigenfunctions, furthermore, form a complete ortho-normal system in X, provided with the inner product induced by $B_t=B_t(·,·)$. Hence it holds that for any $v\in V$ and $i,j=1,2,\cdots$.

The abstract theory developed in [9] is stated as follows. First, if holds for fixed $t\in I$, we obtain at this t, where $j=1,2,\cdots$ are arbitrary (Theorem 8 of [9]). Second, if there exist bilinear forms ${\dot{A}}_t:V\times V\to \mathbb{R}$ and ${\dot{B}}_t:X\times X\to \mathbb{R}$ for any $t\in I$ such that and if it holds that for fixed $t\in I$, we obtain the existence of the unilateral derivatives at this t, where $j=1,2,\cdots$ are arbitrary (Theorem 12 of [9]). If inequalities (5) are valid to any $t\in I$ and it also holds that for the specified t, furthermore, the above unilateral derivatives satisfy

(Theorem 13 of [9]).

We assume, furthermore, for some $k,m=1,2,\cdots$, under the agreement of ${\lambda }_0\left(t\right)=-\infty$. Assume, also that (5) and (7) hold for any $t\in I$. Then, there exists a family of $C^1$ curves denoted by ${\tilde{C}}_i$, $k\le i\le k+m-1$, made by a rearrangement of at most countably many times in I, where

(Theorem 3, Theorem 14 of [9]).

Third, if we have the other bilinear forms ${\ddot{A}}_t:V\times V\to \mathbb{R}$ and ${\ddot{B}}_t:X\times X\to \mathbb{R}$ satisfying for any $t\in I$, and for the fixed t, then there are for this t, where $j=1,2,\cdots$ are arbitrary (Remark 12 of [9]). These ${\ddot{\lambda }}_j^{\pm }\left(t\right)$, furthermore, satisfy if inequalities (10) are valid to any $t\in I$ and it holds that for the above specified t (Theorem 23 of [9]). Furthermore, if (10) and (11) hold for any $t\in I$ and if it holds that (8), then the above described family of curves ${\tilde{C}}_i$, $k\le i\le k+m-1$, are $C^2$ (Theorem 24 of [9]).

Finally, these derivatives ${\dot{\lambda }}_j^{\pm }\left(t\right)$ and ${\ddot{\lambda }}_j^{\pm }\left(t\right)$ for $k\le j\le k+m-1$ are characterized as the eigenvalues of associated eigenvalue problems on the m-dimensional space, $\langle u_j\left(t\right)\mid k\le j\le k+m-1\rangle$ (Theorem 12 and Theorem 15 of [9]). Henceforth, we assume (1), (2), (4), (5), (7), (9), (10), and (11) for any $t\in I$.

If $m=1$ holds in (8), for example, we obtain and where $\lambda ={\lambda }_k\left(t\right)$, ${\varphi }_t=u_k\left(t\right)$, and $C_t^{\lambda }=A_t-\lambda B_t$. Here, $w=\gamma \left(u\right)\in V_1$ is defined for $u\in V$ by where $V_1=PV$, $P=I-R$, and $R:X\to V_0=\langle {\varphi }_t\rangle$ is the orthogonal projection, and

Thus we obtain the following theorem because the bilinear form $C_t^{\lambda }$ is non-negative definite on $V\times V$ if $\lambda ={\lambda }_1$.

Since

Theorem 1 implies the following result.

Harmonic convexity of the first eigenvalue, inequality (16), was noticed by [3] for the Dirichlet problem of Laplacian under the conformal deformations of the domain in two space dimension. Here we calculate the values $\dot{A}$, $\ddot{A}$, $\dot{B}$, $\ddot{B}$ and the validity of (15) under general setting of the deformations of the domain, and then turn to the dynamical and the conformal deformations. Taking preliminaries in §2, thus, we show the results on dynamical and conformal deformations in §3 (Theorem 5 and Theorem 6) and §4 (Theorem 8 and Theorem 9), respectively. As applications, we show several new inequalities on the first eigenvalue of the two-dimensional problem.

Let $\Omega$ be a bounded Lipschitz domain in n-dimensional Euclidean space ${\mathbb{R}}^n$ for $n\ge 2$. Suppose that its boundary $\partial \Omega$ is divided into two relatively open disconnected sets ${\gamma }_0$ and ${\gamma }_1$, satisfying

We study the eigenvalue problem of the Laplacian with mixed boundary condition, where and $\nu$ denotes the outer unit normal vector on $\partial \Omega$. This problem takes the weak form, finding u satisfying defined for and where

This V is a closed subspace of $H^1(\Omega )$ under the norm

The above reduction of (18) to (19)-(22) is justified via the trace operator to the boundary since $\Omega$ is a bounded Lipschitz domain (Theorem 2 of [8]).

To confirm the well-posedness of (3), we note, first, that if ${\gamma }_0\ne \varnothing$, there is a coercivity of $A:V\times V\to \mathbb{R}$, which means the existence of $\delta >0$ such that

If ${\gamma }_0=\varnothing$ we replace A by $A+B$, denoted by $\tilde{A}$. Then this $\tilde{A}:V\times V\to \mathbb{R}$ is coercive, and the eigenvalue problem is equivalent to (3) by $\tilde{\lambda }=\lambda +1$. Hence we can assume (1), using this reduction if it is necessary.

If the bounded domain $\Omega \subset {\mathbf{R}}^n$ is provided with the cone property, the inclusion $H^1(\Omega )\hookrightarrow L^2(\Omega )$ is compact by Rellich-Kondrachov’s theorem [1]. Thus there is a sequence of eigenvalues to (3), denoted by

The associated eigenfunctions, $u_1,u_2,\cdots$, furthermore, form a complete ortho-normal system in X, provided with the inner product induced by $B=B(·,·)$:

The j-th eigenvalue of (3) is given by the mini-max principle where is the Rayleigh quotient, and $\left\{L_j\right\}$ and $\left\{W_j\right\}$ denote the families of all subspaces of V with dimension and codimension j and $j-1$, respectively.

The following well-known fact is valid without the smoothness of $\partial \Omega$. The proof given in Appendix A for completeness.

Coming back to the Lipschitz bounded domain $\Omega$, we introduce its deformation as follows. Let be a family of bi-Lipschitz homeomorphisms. We assume that $T_{t}x$ is continuous in t uniformly in $x\in \Omega$, and continue to use the following definition as in [9].

Putting in (18), we introduce the other eigenvalue problem which is reduced to finding for

Let ${\lambda }_j\left(t\right)$ be the j-th eigenvalue of the eigenvalue problem (27). Then Lemma 7 of [9] ensures that the eigenvalue problem (28)-(29) is reduced to by the transformation of variables $y=T_{t}x$, where for V and X defined by (22), and

Recall that $\Omega \subset {\mathbb{R}}^n$ is a bounded Lipshitz domain, and let $T_t:\Omega \to {\Omega }_t=T_t\Omega$, $t\in I=(-{\epsilon }_0,{\epsilon }_0)$ be a family of twice continuously differentiable bi-Lispchitz transformations. Then the abstract theory described in §1 is applicable with and

Henceforth, we write for the symmetric matrix $G\in M_n\left(\mathbb{R}\right)$, where ${·}^T$ denotes the transpose of vectors or matrices. The unit matrix is denoted by $E\in M_n\left(\mathbb{R}\right)$, and for $G=\left(g_{ij}\right)$ and $F=\left(f_{ij}\right)\in M_n\left(\mathbf{R}\right)$. Let, furthermore, $I:\Omega \to \Omega$ be the identity mapping.

Proof: First, since uniformly on $\Omega$, it holds that uniformly on $\Omega$, which implies uniformly on $\Omega$ (Lemma 5 and Lemma 6 of [8]).

Second, we have using the standard $L^2$ inner product $(,)$, and therefore, it follows that and

By these equalities, finally, we obtain and and hence the conclusion.

We continue to suppose that $\Omega \subset {\mathbf{R}}^n$ is a bounded Lipschitz domain. Here we study the dynamical deformation of domains introduced by [7].

To this end, we take a Lipschitz continuous vector field defined on a neighbourhood of $\overline{\Omega }$, denoted by $v=v\left(x\right)$. Then the transformation $T_t:\Omega \to {\Omega }_t=T_t(\Omega )$ is made by $T_{t}x=X\left(t\right)$, where $X=X\left(t\right)$, $\left|t\right|\ll 1$, is the solution to

Then we have the group property, and therefore, the formulae derived for $t=0$ are shifted to $t=s$.

If v is a $C^{1,1}$ vector field, furthermore, this $T_t:\Omega \to {\Omega }_t$ is twice continuously differentiable, and it holds that (33) with because of

Then we obtain the following lemma.

Proof: Writing $v=\left(v^j\right)$, we obtain and the proof is complete. □

Here we take two categories that the vector fields are solenoidal and gradient. In the first category, we assume $\nabla ·v=0$ everywhere. Then it holds that

Proof: We recall (34). By the assumption we obtain $\nabla ·S=0$ and by Lemma 1, which implies $\dot{B}=\ddot{B}=0$, and (36) by Theorem 4. □

In the second category that the vector field is a gradient of a scalar field we obtain the following theorem.

Proof: In this case, we obtain $S=\nabla \mu$ and

Then it holds that

It also holds that because is valid to the vector field a.

Since is valid by Lemma 1, and hence

Next, we obtain and hence

Finally, we divide $\ddot{A}$ as in for

Here we have

Equality (39) now implies for $R=\left(R^i\right)$, and hence

Thus we obtain

It holds also that

Finally, we have