Georgian Mathematical Journal Volume 9 (2002), Number 3, 449–459

ON A GENERALIZATION OF THE DIRICHLET INTEGRAL

T. CHANTLADZE, N. KANDELAKI, A. LOMTATIDZE, AND D. UGULAVA

Abstract. Using the theory of spline functions, we investigate the problem of minimization of a generalized Dirichlet integral

F_{λ}(u) = ∫_{Ω} (λ^{2} + ∑_{i=1}^{n} u_{x_{i}}^{2})^{p/2}, 1 < p < ∞,

where Ω is a bounded domain of an n-dimensional Euclidean space R_{n}, λ ≥ 0 is a fixed number, and u_{x_{i}} is a generalized according to Sobolev with respect to x_{i} derivative of the function u defined on Ω. Minimization is realized with respect to the functions u whose boundary values on Γ form a preassigned function, and for them F_{λ}(u) is finite.

2000 Mathematics Subject Classification: 35J40.

Key words and phrases: Dirichlet integral, spline functions, Sobolev space, generalized derivative, l-Laplacian.

Let X and Z be linear normed spaces and Y be a Banach space possessing properties E and U. Property E means that in any closed subspace there exists an element g, least deviating from the given one y ∈ Y. Property U shows that such an element g is unique. It is known that the linear normed space possesses property E if and only if it is reflexive, and property U if and only if it is strictly convex (i.e., when the norm of a midpoint of the segment connecting two different points on the unit sphere, is less than 1). Suppose that t and a are continuous linear mappings from X into Y and from X into Z, respectively. For a fixed element z ∈ Im a let us consider its pre-image in X, i.e., a set I_{z} = {x ∈ X : a(x) = z} and then in Y, i.e., a set Δ_{z} = t(I_{z}). This is the set of those elements in Y which have the form y = t(x), where x ∈ I_{z}. Find ∥t(x)∥_{Y} when x ∈ I_{z}. If this inf is attained on some element σ from the set I_{z}, then it is called a spline corresponding to t, a and z. By the definition,

∥t(σ)∥_{Y} = inf_{X∈I_{z}} ∥t(X)∥_{Y} .  (1)

Such a definition of a spline was introduced first for Hilbert spaces X, Y, Z ([1], Ch. 4, §4.4). The above-given definition of the spline is given in [2].

In the sequel, instead of Ker(·) the operator kernel will be denoted by N(·).

Theorem 1. The spline corresponding to t, a and z exists if the set t(N(a)) is closed in Y. It is unique if N(a) ∩ N(t) = {0}.

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450 T. CHANTLADZE, N. KANDELAKI, A. LOMTATIDZE, AND D. UGULAVA

Proof. Clearly, the set Δ_{z} = t(I_{z}) is a shift of the subspace Δ_{0} = t(I_{0}) = t(N(a)) by the element t(x_{0}), where x_{0} is an arbitrary fixed element from I_{z}. By the condition, any subspace of the space Y possesses properties E and U. Then the shifts of the subspace possess the same properties. (Indeed, let x ∈ Y, x ∉ Δ_{z} and let x_{0} be an arbitrary element from Δ_{z}. For x−x_{0} in Δ_{0} there exists a best approximation element of g_{0}, i.e., ∥x−x_{0}−g_{0}∥_{Y} = inf_{g∈Δ_{0}} ∥x−x_{0}−g∥_{Y} = inf_{g∈Δ_{0}} ∥x−(x_{0} +g)∥_{Y} = inf_{_{g}∈Δ_{z}} ∥x−g̃∥_{Y}, i.e., g̃_{0} = x_{0} +g_{0} is the element of the best approximation in Δ_{z}. Analogously, we can see that g̃_{0} is unique.) Since, by the condition, the set Δ_{0} = t(I_{0}) is closed, it is a subspace, and hence its shift Δ_{z} likewise possesses properties E and U. Let \overline{y} be a best approximation element of zero in Δ_{z}, i.e., ∥\overline{y}∥_{Y} = inf_{y∈Δ_{z}} ∥y∥_{Y}. Thus we have proved the existence of the spline.

Let us prove its uniqueness under the condition N(a) ∩ N(t) = {0}. Suppose that there exist two splines σ and σ̃ ∈ I_{z}, a(σ) = a(σ̃) = z, a(σ − σ̃) = 0, i.e., σ − σ̃ ∈ I_{0} = N(a). Since \overline{y} is a best approximation unique element, t(σ) = t(σ̃) = y, t(σ − σ̃) = 0, i.e., σ − σ̃ ∈ N(a) ∩ N(t) = {0} and σ = σ̃.

Remark. Let y_{0} be an arbitrary fixed element, and instead of problem (1) let us consider a somewhat modified problem. We are to find that inf_{x∈I_{z}} ∥y_{0} + t(x)∥_{Y}. If the conditions of Theorem 1 are fulfilled, then there exists an element σ̃ ∈ I_{z} for which

∥y_{0} + t(σ̃)∥_{Y} = inf_{x∈I_{z}} ∥y_{0} + t(x)∥_{Y} .

Now we need establish the sufficient condition for the set t(N(a)) from Y to be closed.

Lemma 1. If X is a Banach space, Im t is closed in Y, and N(t) is finitedimensional, then t(N(a)) is a closed subspace in Y.

Proof. Since N(t) is finite-dimensional, it has a complement, i.e., there exists a subspace N(t)^{⊥} in X such that N(t)∩N(t)^{⊥} = {0}, X = N(t)⊕N(t)^{⊥}. Every element x ∈ X is represented uniquely as x = u + v, u ∈ N(t), v ∈ N(t)^{⊥}. Denote by t̃ the restriction of the mapping t on N(t). It is clear that Im t = Im t̃, and t̃ is a continuous one-to-one mapping acting from N(t)^{⊥} to a closed subspace of the space Y. According to the Banach theorem on the continuity of an inverse mapping, s = t^{−1} is a continuous mapping. Denote by N_{a}(t) the projection of N(a) onto N(t)^{⊥}. We have

t(N(a)) = t(N_{a}(t)) = t̃(N_{a}(t))
= s^{−1}(N_{a}(t)) = s^{−1}((N(a) + N(t)) ∩ N(t)^{⊥}).  (2)

Here we have taken into account the fact that N_{a}(t) = ((N(a)+N(t))∩N(t))^{⊥}. Indeed, if x ∈ N_{a}(t), then x ∈ N(t)^{⊥}, i.e., x is the projection of some element x_{1} ∈ N(a) onto N(t) and x = x_{1} − x_{2}, where x_{2} ∈ N(t). This implies that x ∈ N(a) + N(t) and hence x ∈ (N(a) + N(t)) ∩ N(t)^{⊥}. Conversely, if x ∈ (N(a) + N(t)) ∩ N(t)^{⊥}, then x = x_{1} + x_{2}, where x_{1} ∈ N(a), x_{2} ∈ N(t). Next, x_{1} = x − x_{2}, but x belongs to N(t)^{⊥} and hence it is the projection of x_{1} onto N(t)^{⊥}, i.e., x ∈ N_{a}(t). N(a) is closed, and since N(t) is finite-dimensional, N(a) + N(t) is likewise closed. Indeed, let the sequence x_{n} ∈ N(a) + N(t), n = 1, 2, …, converge in X. This sequence can be written as follows:

x_{n} = \overline{x}_{n} + ∑_{k=1}^{r} α_{n}^{(k)}t_{k},  (3)

where \overline{x}_{n} ∈ N(a), and t_{1}, …, t_{r} are linear independent elements from N(t) such that a(t_{1}), …, a(t_{r}) are also linear independent in Z. (Indeed, let {t_{k}}_{1≤k≤m} be a basis in N(t) such that a(t_{1}), …, a(t_{r}), r ≤ m, is a maximal linear independent subset of the set {a(t_{k})}_{1≤k≤m}. Then for t_{j}, j = r + 1, …, m, we have a(t_{j}) = ∑_{k=1}^{r} β_{k}^{(j)}a(t_{k}); and t_{j} − ∑_{k=1}^{r} β_{k}^{(j)}t_{k} = ξ_{j} ∈ N(a), or t_{j} = ∑_{k=1}^{r} β_{k}^{(j)}t_{k} +ξ_{j}, β_{k}^{(j)} ∈ R, ξ_{j} ∈ N(a), r + 1 ≤ j ≤ m. But, by the definition, x_{n} = x̃_{n} + ∑_{k=1}^{m} γ_{n}^{(k)}t_{k}, x̃_{n} ∈ N(a), γ_{n}^{(k)} ∈ R. Taking into account all the above-said, we can see that (3) is valid). By virtue of the convergence of (3) and the continuity of the mapping a, the sequence a(x_{n}) = ∑_{k=1}^{r} α_{n}^{(k)}a(t_{k}) converges as well.

Thus we conclude that for all 1 ≤ k ≤ r the number of sequences α_{n}^{(k)} converge. Further, according to (3), there exists lim \overline{x}_{n} = \overline{x} ∈ N(a), and if lim x_{n} = x, then x ∈ N(a) + N(t). Hence N(a) + N(t) is a closed set. N(t) is closed, and since s^{−1} is a continuous mapping, from (2) it follows that the set t(N(a)) is closed in Y.

Let us realize the above-described construction for the following case. Let Ω be a simply-connected bounded domain of an n-dimensional Euclidean space R_{n} for which Sobolev embedding theorems (see, e.g., [3], pp. 279–280) are valid. For instance, they are valid if Ω is representable in the form of a union of a finite number of star-shaped domains.

Let X = W_{p}^{1}(Ω) be a set of integrable on Ω in the p-th degree, 1 < p < ∞, functions u(x) = u(x_{1}, …, x_{n}) having the integrable in the p-th degree generalized, according to Sobolev, partial derivatives of first order u_{x_{i}}, 1 ≤ i ≤ n. The set of such functions with the norm

∥u∥_{W^{1}_{p}(Ω)} = {∫_{Ω} (u_{x_{1}}^{2} + ⋯ + u_{x_{n}}^{2})^{p/2}dσ}^{1/p}+ {∫_{Ω} |u|^{p}dσ}^{1/p}, dσ =dx_{1}, …, dx_{n},

is the Banach space, or more precisely, the Sobolev space (its different equivalent norms are well-known (see, e.g., [3]). In the capacity of the space Y we take the Cartesian product of n samples of the space L_{p}(Ω) and of one-dimensional space R, i.e., Y = R × L_{p}(Ω) × ⋯ × L_{p}(Ω) = R × ∏_{n} L_{p}(Ω). Introduce in Y the norm of the element (λ, u_{1}, …, u_{n}) as follows:

∥(λ, u_{1}, …, u_{n})∥_{Y} = ∥(λ^{2} + u_{1}^{2} + ⋯ + u_{n}^{2})^{1/2}∥_{Lp(Ω)}
= { ∫_{Ω} (λ^{2} + ∑_{i=1}^{n} u_{i})^{p/2}dv}^{1/p}.  (4)

It is known that L_{p}(Ω) is a reflexive and strictly normed space for 1 < p < ∞. The same will be Y with norm (4). Let us prove that the space Y is strictly convex. To this end, we take in Y different points u = (λ_{1}, u_{1}, …, u_{n}) and v = (λ_{2}, v_{1}, …, v_{n}) with norms, equal to unity, and prove that ∥u + v∥_{Y} < 2.

By means of the inequality

\sqrt{(λ_{1} + λ_{2})^{2} + (u_{1} + v_{1})^{2} + ⋯ + (u_{n} + v_{n})^{2}}
≤ \sqrt{λ_{1}^{2} + u_{1}^{2} + ⋯ + u_{n}^{2}} + \sqrt{λ_{2}^{2} + v_{1}^{2} + ⋯ + v_{n}^{2}}  (5)

we get

∥u + v∥_{Y} = { ∫_{Ω} ((λ_{1} + λ_{2})^{2} + (u_{1} + v_{1})^{2} + ⋯ + (u_{n} + v_{n})^{2})^{p/2}dσ}^{1/p}

≤ \{ ∫_{Ω} (\sqrt{λ_{1}^{2} + u_{1}^{2} + ⋯ + u_{n}^{2}} + \sqrt{λ_{2} + v_{1}^{2} + ⋯ + v_{n}^{2}})^{p}dσ\}^{1/p}

=^{def} { ∫_{Ω} (f_{1} + f_{2})^{p}dσ}^{1/p},  (6)

where

f_{1} = \sqrt{λ_{1}^{2} + u_{1}^{2} + ⋯ + u_{n}^{2}}, f_{2} = \sqrt{λ_{2}^{2} + v_{1}^{2} + ⋯ + v_{n}^{2}},
∥f_{1}∥_{Lp(Ω)} = ∥f_{2}∥_{Lp(Ω)} = ∥u∥_{Y} = ∥v∥_{Y} = 1.

Consider the following cases:

(1) f_{1} ≠ f_{2}. By virtue of the strict convexity of the space L_{p}(Ω) for 1 < p < ∞, from (6) we find that

∥u + v∥_{Y} < ∥f_{1}∥_{Lp(Ω)} + ∥f_{2}∥_{Lp(Ω)} = ∥u∥_{Y} + ∥v∥_{Y} = 2;

(2) f_{1} = f_{2}, i.e., λ_{1}^{2} +u_{1}^{2} +⋯+u_{n}^{2} = λ_{2}^{2} +v_{1}^{2} +⋯+v_{n}^{2} a.e. on Ω. Suppose first that at least one of the numbers λ_{1} and λ_{2} is not equal to zero. The equality in inequality (5) is obtained, when the vectors u and v are proportional a.e., and since the first coordinates are the numbers, the coefficient of proportionality is a positive number C, i.e., v = Cu. But ∥u∥ = ∥v∥ = 1 and hence C = 1, v = u which is impossible by our supposition.

If λ_{1} = λ_{2} = 0, then the equality a.e. in (5) is obtained if a.e. v = C(x)u, where C(x) > 0 for almost all x ∈ Ω. In this case u = 0 a.e. in Ω if and only if v = 0 a.e. in Ω. But |v| = |u| a.e., and hence C(x) = 1 a.e. in Ω. We obtain u = v and this contradicts our supposition. Thus the strict convexity of the space Y is proved.

In the sequel, we shall need some of the Sobolev embedding theorems on a structure of traces for elements from Sobolev spaces on manifolds of lesser dimensions. As such a manifold we take the boundary Γ of the domain Ω which is required to consist of a finite number of surfaces of the class C^{1}. By the Sobolev embedding theorems, for p > n the function f from the space W^{1}p(Ω) is continuous everywhere in Ω, including the boundary Γ. Moreover,

max_{x∈Γ} |f(x)| ≤ c∥f∥_{W^{1}_{p}(Ω)},  (7)

where c is the constant depending on Γ. If 1 < p ≤ n, the function f ∈ W_{p}^{1}(Ω) has a trace on Γ, which belongs to L_{q}(Γ), where q < (n − 1)p/(n − p) and

∥f∥_{Lq(Γ)} ≤ c∥f∥_{W^{1}_{p}(Ω)}.  (8)

Since (n − 1)p/(n − p) > p, for 1 < p ≤ n the trace of elements of the space W_{p}^{1}(Ω) on Γ belongs to L_{p}(Γ) (the fact that f|_{Γ} is the trace of the element f ∈ W_{p}^{1}(Ω) on Γ means that in the class of equivalent functions f we can find a function which possesses limit values in the ordinary, or in the L_{p} sense. The above-mentioned theorems are proved, for example, in [3] or in [4].

As the Banach space Z we introduce the space of L_{p}(Γ)-functions, integrable on Γ in the p-th degree. Denote by a a linear operator from X to Z which to the element u ∈ X puts into correspondence its trace on the boundary Γ of the domain Ω. Inequalities (7) and (8) imply that a is a linear continuous operator. The operator t, acting from X to Y, can be defined by the formula

t(u) = (0, u_{x_{1}}, …, u_{x_{n}}).

Clearly,

∥t(u)∥_{Y} = { ∫_{Ω} ( ∑_{i=1}^{n} u_{x_{i}}^{2})^{p/2}dv}^{1/p}.

The right-hand side of the latter equality is the summand of the norm ∥u∥_{X}, and hence t is a linear continuous operator.

In the space Z we fix an element f ∈ Im a and denote by I_{f} a set of elements u of the space X whose trace on Γ is f, i.e., a(u) = f, Λ = (λ, 0, …, 0) belongs to Y, and the set Λ + t(u) = (λ, u_{x_{1}}, …, u_{x_{n}}) for all possible u ∈ I_{f} is a shift of the set t(N(a)) onto the element (λ, v_{x_{1}}, …, v_{x_{n}}), where v is some fixed element from I_{f}.

Consider the functional

F_{λ}(u) = { ∫_{Ω} (λ^{2} + ∑_{i=1}^{n} u_{x_{i}}^{2})^{p/2}dσ}^{1/p} = ∥Λ + t(u)∥_{Y}  (9)

defined on I_{f} and find its inf with respect to all u ∈ I_{f}. By Theorem 1, the closure of the set t(N(a)) in Y and the equality N(a) ∩ N(t) = {0} guarantee the existence and uniqueness of the minimizing element ũ ∈ I_{f}. To prove the closure of the set t(N(a)), it is sufficient, following Lemma 1, to prove finite dimensionality of N(t) and closure of Im t in Y. N(t) is the set of functions from X whose all generalized partial derivatives of first order are zeros. This means that N(t) is a one-dimensional space of constants. It remains to prove the closure Im t. Let u^{(m)} be a sequence in X such that all the sequences u_{x_{i}}^{(m)}, 1 ≤ i ≤ n, composed of generalized partial derivatives of u^{(m)} converge in L_{p}(Ω) to the functions α_{i}. We have to prove that the vector (α_{1}, …, α_{n}) is the generalized gradient of a function u ∈ X, i.e., u_{x_{i}} = α_{i}, 1 ≤ i ≤ n. For 1 ≤ i, j ≤ n denote by (u^{(m)})_{x_{i}x_{j}} the derivative with respect to x_{j} of the function (u^{(m)})_{x_{i}} in a sense of the theory of generalized functions. Then for any finite in Ω function f we have

(u^{(m)})_{x_{i}x_{j}}(f) = \frac{∂u^{(m)}}{∂x_{i}} (\frac{∂f}{∂x_{j}} ) → ∫_{Ω} α_{i} \frac{∂f}{∂x_{j}} dσ,

(u^{(m)})_{x_{j}x_{i}}(f) = \frac{∂u^{(m)}}{∂x_{j}} (\frac{∂f}{∂x_{i}} ) → ∫_{Ω} α_{j} \frac{∂f}{∂x_{i}} dσ.

The left-hand sides of these relations are equal to u^{(m)}(f_{x_{i}x_{j}}) (elements u^{(m)} ∈ X are considered as regular generalized functions). Hence their right-hand sides are likewise equal:

∫_{Ω} α_{i} ∂f/∂x_{j}dσ = ∫_{Ω} α_{j}∂f/∂x_{i}dσ,

i.e., (α_{j})_{x_{i}} = (α_{i})_{x_{j}}, where the derivatives are considered in a sense of the theory of generalized functions. It is known ([3], p. 160; [5], Ch. 5, 5.9.3, 5.11.1) that in the case, where Ω is simply connected, there exists a locally integrable in Ω function u whose generalized partial derivatives u_{x_{i}}, 1 ≤ i ≤ n, belong to L_{p}(Ω), and grad u = (α_{1}, …, α_{n}). But by the Sobolev embedding theorem ([3], pp. 280–282), such a function u belongs to L_{p}(Ω), and hence u ∈ X, i.e., Im t is closed in Y. The condition N(a) ∩ N(t) = {0} can be easily verified. Thus we have proved the following

Theorem 2. Let Ω be a bounded simply connected domain from R_{n}, whose boundary Γ consists of a finite number of surfaces of the class C^{1}, f be the defined on Γ function for which the set I_{f} = {u ∈ W_{p}^{1}(Ω) : u|_{Γ} = f} is nonempty, and let λ be an arbitrary fixed real number. Then for the functional F_{λ}, defined by formula (9), inf_{u∈I_{f}} F_{λ}(u) is obtained on some element u ∈ I_{f}.

Remark 1. Theorem 2 is also valid for the domain which is divided by means of surfaces of the class C^{1} into a finite number of simply connected domains. Indeed, the functions u ∈ I_{f} will have, according to the above-mentioned Sobolev theorems, on such surfaces a trace belonging to the corresponding space L_{p}, and the problem of minimization of the functional F_{λ} can be considered in the obtained simply connected domains. Using minimizing elements, we can construct a minimizing function in the initial domain.

Remark 2. Using the method described above, we can investigate a functional of more general type such as

F_{λ,s}(u) = { ∫_{Ω} (λ^{2} + Du)^{p/2}dσ}^{1/p},  (10)

where

Du = ∑_{s_{1}+···+s_{n}=s} c(s_{1}, …, s_{n})(\frac{∂^{s}u}{∂x_{1}^{s_{1}} … ∂x_{n}^{s_{n}}} )^{2}, c(s_{1}, …, s_{n}) > 0,

and λ is a fixed function belonging to L_{p}(Ω), while the derivatives are understood in the Sobolev sense. Not going into details, we note that for functional (10) we obtain a theorem analogous to Theorem 2. Here we restrict ourselves to case (9).

For the element u minimizing functional (9) we derive the so-called Euler’s generalized equation. To this end, we consider some function η belonging to D(Ω), a space of finite, infinitely differentiable in Ω function, and for the real numerical parameter α we compose the set {u + αη}. Let p = l + 1, l > 0. The function u minimizes the functional

I_{λ}(u) = ∫_{Ω} (λ^{2} + ∑_{i=1}^{n} u_{x_{i}}^{2})^{(l+1)/2}dσ,

and since the trace of the function u + αη belonging to W_{l+1}(Ω) on Γ is f, for any α ∈ R we have

I_{λ}(u + αη) ≥ I_{λ}(u).

Let us prove that

I_{λ}(u + αη) = ∫_{Ω} (λ^{2} + ∑_{i=1}^{n}(u_{x_{i}} + αη_{x_{i}})^{2})^{(l+1)/2}dσ  (11)

has a derivative with respect to α for any α ∈ R, and I′_{λ} can be calculated by differentiation under the integral sign. For this we write the expression

\frac{ΔI_{λ}}{Δα} = \frac{I_{λ}(u + (α + Δα))η − I_{λ}(u + αη)}{Δα}

(Δα is an increment of α) and transform it by means of the Lagrange mean value theorem. We obtain

\frac{ΔI_{λ}}{Δα} = (l + 1) ∫_{Ω} (λ^{2} + ∑_{i=1}^{n}(u_{x_{i}} + α_{θ}η_{x_{i}})^{2})^{(l−1)/2}

×[ ∑_{i=1}^{n}(u_{x_{i}} + α_{θ}η_{x_{i}})η_{x_{i}}]dσ,  (12)

where α_{θ} = α + θ(u_{x_{1}}, …, u_{x_{n}}, α), 0 < |θ(u_{x_{1}}, …, u_{x_{n}}, α)| < |Δα|.

By the Cauchy–Bunjakovsky inequality we find that the integrand is estimated from the above by the expressions

[λ^{2} + ∑_{i=1}^{n}(u_{x_{i}} + α_{θ}η_{x_{i}})^{2}]^{l/2}( ∑_{i=1}^{n} η_{x_{i}}^{2})^{1/2}

and

max {1, [λ^{2} + ∑_{i=1}^{n}(u_{x_{i}} + α_{θ}η_{x_{i}})^{2}]^{l+1/2}}( ∑_{i=1}^{n} η_{x_{i}}^{2})^{1/2}.

But

λ^{2} + ∑_{i=1}^{n}(u_{x_{i}} + α_{θ}η_{x_{i}})^{2} ≤ (λ^{2} + ∑_{i=1}^{n} u_{x_{i}}^{2})(1 + |α_{θ}|) + (|α_{θ}| + α_{θ}^{2}) ∑_{i=1}^{n} η_{x_{i}}^{2},

and u ∈ W_{l+1}. Therefore we can easily see that the integrand in (12) has a majorant belonging to L_{1}(Ω). By the Lebesgue theorem, we can pass in (12) to the limit under the integral sign. Thus from (11) we find that

I′_{λ}(u + αη) = (l + 1) ∫_{Ω} (λ^{2} + ∑_{i=1}^{n}(u_{x_{i}} + αη_{x_{i}})^{2})^{(l−1)/2}[ ∑_{i=1}^{n}(u_{x_{i}} + αη_{x_{i}})η_{x_{i}}]dσ.

Since α = 0 is the minimum point of the function I_{λ}(u + αη), we have that I′_{λ}(u) = 0, and hence the function u, minimizing I_{λ}(u), satisfies the generalized Euler equation

∫_{Ω} (λ^{2} + ∑_{i=1}^{n} u_{x_{i}}^{2})^{(l−1)/2}( ∑_{i=1}^{n} u_{x_{i}}η_{x_{i}})dσ = 0  (13)

for any function η ∈ D(Ω). For λ = 0 we get the equation

∫_{Ω} | grad u|^{l−1}( ∑_{i=1}^{n} u_{x_{i}}η_{x_{i}})dσ = 0.  (14)

It is seen from (14) that u is a solution of the equation

Δ_{l}(u) =^{def} div (| grad u|^{l−1} grad u) = ∑_{i=1}^{n} \frac{∂}{∂x_{i}} (| grad u|^{l−1}u_{x_{i}}) = 0,  (15)

where partial derivatives ∂/∂x_{i} are understood in a sense of the theory of generalized functions. The solution of equation (13) satisfies in the same sense equation (14), where grad u is replaced by an expression of the type (λ^{2} +

∑_{i=1}^{n} u_{x_{i}}^{2})^{1/2}. Equation (14) is known in the literature as the l-Laplace equation, and the corresponding operator Δ_{l} is called the l-Laplacian ([8]). For λ = 0, l = 1 the functional F_{λ}, defined by formula (9), coincides with the known Dirichlet integral, and Δ_{1} is the known Laplace operator.

Let now the function u belonging to the space W_{l+1}^{1}(Ω), l > 0, satisfy in Ω the equation

{Δ_{l}}_{λ}u = div ((λ^{2} + | grad u|^{2})^{(l−1)/2} grad u) = 0  (16)

in a sense of the theory of generalized functions. Suppose that the trace of the function u on the boundary Γ of the domain Ω is the function f ∈ L_{l+1}(Γ). Let us prove that in such a case a solution u ∈ W_{l+1}^{1}(Ω) of equation (16) minimizes functional (9) in the class I_{f}. As above, I_{f} denotes the set of functions from W_{l+1}^{1}(Ω) whose trace on Γ is f. It is clear that the solution u of equation (16) satisfies equation (13) for any function η ∈ D(Ω). Note first that (13) is also satisfied for any function η ∈ W_{l+1}^{1}(Ω) whose trace on Γ is zero. We

denote the set of such functions by W^{∘} _{l+1}^{1}(Ω). Indeed, let η be an arbitrary

function from W^{∘} _{l+1}^{1}(Ω). Then there exists in Ω a sequence η_{n} ∈ D(Ω) such that ∥η − η_{n}∥_{W_{l+1}}^{1}(Ω) → 0 as n → ∞. For l = 1 this proposition is known (see, e.g., [6], p. 119). The proof is analogous for any l > 0. The integral in the left-hand side of (10) exists if u, η ∈ W_{l+1}^{1}(Ω). Indeed, from the Hölder inequality we obtain

∫_{Ω} (λ^{2} + | grad u|^{2})^{(l−1)/2}( ∑_{i=1}^{n} u_{x_{i}}η_{x_{i}})dσ

≤ ∫_{Ω} (λ^{2} + | grad u|^{2})^{(l−1)/2}| grad u∥ grad η|dσ

≤ ∫_{Ω} (λ^{2} + | grad u|^{2})^{l/2}| grad η|dσ

≤ {∫_{Ω} (λ^{2} + | grad u|^{2})^{(l+1)/2}σ}^{(l+1)/l}{∫_{Ω} | grad u|^{(l+1)}dσ}^{1/(l+1)} <∞. (17)

Since (10) is valid for η_{n} ∈ D(Ω), estimate (17) shows that (10) is now valid

for any function η ∈ W^{∘} _{l+1}^{1}(Ω).

Along with the solution u of equation (13), we consider the set {u + αη},

α ∈ R, η ∈ W^{∘} _{l+1}^{1}(Ω). Since (u + αη)|_{Γ} = f, u + αη ∈ I_{f}. Then (9) implies that

F_{λ}(u + αη) = ∫_{Ω} (λ^{2} + | grad(u + αη)|^{2})^{(l+1)/2}dσ.  (18)

Denote now (λ^{2} + | grad(u)|^{2})^{(l+1)/2} by ϕ_{l}(α) and prove that for any l > 0 and α ∈ R the estimate

ϕ_{l}(α) ≥ ϕ_{l}(0) + αϕ_{l}(0)  (19)

is valid. To this end, we prove the following

Lemma 2. If α ∈ R and a = (a_{1}, …, a_{n}), b = (b_{1}, …, b_{n}) are the points from R_{n}, then the inequality

(λ^{2} + ∑_{i=1}^{n}(a_{i} + b_{i})^{2})^{(l+1)/2}

≥ (λ^{2} + ∑_{i=1}^{n} a_{i}^{2})^{(l+1)/2} + (l + 1)(λ^{2} + ∑_{i=1}^{n} a_{i}^{2})^{(l−1)/2}( ∑_{i=1}^{n} a_{i}b_{i})  (20)

is valid for any l > 0.

Proof. Let t and s be vectors from R_{k}, and the length s be equal to unity. Then |s + t| ≥ |1 + (t, s)|, where (t, s) is the scalar product of the vectors t and s. Indeed, if θ is the angle between the vectors t + s and t, then |s + t| ≥ |s + t||s|| cos θ| = |(s + t, s)| = |(s, s) + (t, s)| = |1 + (t, s)|. The (1 + l)-th degree of |1 + (t, s)| can be estimated from below: |1 + (t, s)|^{(l+1)} ≥ 1 + (l + 1)(t, s); this inequality coincides, in fact, with that of [9] (p. 12). Therefore

|s + t|^{(l+1)} ≥ 1 + (l + 1)(t, s).  (21)

Let x be an arbitrary non-zero vector from R_{k}. Substituting s = x/|x| and y = |x|t into (21), we obtain

|x + y|^{(l+1)} ≥ |x|^{(l+1)} + (l + 1)|x|^{(l−1)}(x, y).  (22)

Clearly, the obtained inequality is likewise valid for x = 0. To estimate (20), we have to substitute x = (λ, a_{1}, …, a_{n}) and y = (0, b_{1}, …, b_{n}) into (22).

Applying now Lemma 2 to a_{i} = u_{x_{i}} and b_{i} = α.η_{x_{i}}, we see that estimate (19) is valid. Further, according to (18) we have

F_{λ}(u + η) ≥ F_{λ}(u) + (l + 1) ∫_{Ω} (λ^{2} + | grad u|^{2})^{(l−1)/2}( ∑_{i=1}^{n} u_{x_{i}}η_{x_{i}})dσ,

and by virtue of (13) we make sure that F_{λ}(u + η) ≥ F_{λ}u for any function

η ∈ W^{∘} _{l+1}^{1}(Ω). As we know, inf in (9) is obtained on some element u_{0} from I_{f}.

Take η = u_{0} − u ∈ W^{∘} _{l+1}^{1}(Ω). Then we have F_{λ}(u_{0}) ≥ F_{λ}(u). Since u_{0} is the unique minimizing element, u = u_{0}, i.e., u minimizes the functional F_{λ}. Thus we proved the following

Theorem 3. Let the function f ∈ L_{l+1}(Γ), l > 0, be given on the boundary Γ of the domain Ω (satisfying the above conditions) which is the trace of a function from the space W_{l+1}^{1}(Ω). Then the element u of the class I_{f} which minimizes functional (9) (u exists by Theorem 2) is a solution of equation (13) in a sense of the theory of generalized functions. Conversely, if there exists a solution u of equation (13) belonging to W_{l+1}^{1}(Ω) and having a trace u|_{Γ} = f, then it minimizes functional (9) in the class I_{f} and is unique.

References

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3. S. L. Sobolev, Introduction to the theory of cubature formulas. (Russian) Nauka, Moscow, 1974.

4. V. I. Smirnov, Course on higher mathematics, V. (Russian) FIZMATGIZ, Moscow, 1959.

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6. V. P. Mikhaylov, Partial differential equations. (Russian) Nauka, Moscow, 1983.

7. N. Kandelaki, A. Lomtatidze, T. Chantladze, and D. Ugulava, The p-Laplacian and connected pairs of functions. Mem. Differential Equations Math. Phys. 20(2000), 113– 126.

8. E. F. Beckenbach and R. Bellman, Inequalities, Springer, Berlin etc., 1983.

(Received 30.01.2002)

Authors’ address:

T. Chantladze, N. Kandelaki, and D. Ugulava

N. Muskhelishvili Intitute of Computational Mathematics Georgian Academy of Sciences 8, Akuri St., Tbilisi 380093 Georgia

A. Lomtatidze

Department of Mathematical Analysis

Faculty of Science

Masaryk University

Janáčkovo nám. 2a, 662 95 Brno

Czech Republic

E-mail: bacho@math.muni.cz