Full Text
V. G. Maz’ya
\(p\)-CONDUCTIVITY AND EMBEDDING THEOREMS FOR CERTAIN FUNCTIONAL SPACES INTO THE SPACE \(C\)
(Presented by Academician V. I. Smirnov on 5 V 1961)
MATHEMATICS
\(1^\circ\). In the note \((^1)\) sufficient, and in some cases necessary and sufficient, conditions were formulated for a domain of \(n\)-dimensional space \(E_n\) under which various embedding theorems hold. In the present work new theorems in this direction are given\(*\).
Let \(D\) be an open set of the space \(E_n\). The boundary of an arbitrary set \(E\) will be denoted by \(\Gamma E\).
Definition 1. By the space \(L_p^{(1)}(D)\) \(\bigl(\widetilde L_p^{(1)}(D)\bigr)\) \((p \ge 1)\) we shall mean the closure of the space \(C^{(1)}(D)\) \(\bigl(C^{(1)}(\overline D)\bigr)\) of functions continuously differentiable in \(D,\overline D\), with respect to the norm
\[
\|u\|_{L_p^{(1)}(D)}=\|\operatorname{grad}u\|_{L_p(D)}.
\]
Definition 2. Let \(E\) and \(F\) be subsets of \(D\) such that \(F\) is an open set, \(E\) is a set closed in \(D\)\(**\), and \(E \subset F\). The set
\[
K=F\setminus E
\]
will be called a conductor in \(D\).
Definition 3. Let \(K\) be an arbitrary conductor in \(D\). We shall say that a function \(u(x)\) belongs to the set \(U(K)\) if: 1) \(u(x)\) is continuously differentiable in \(K\); 2) \(u(x)=1\) on the set \(\Gamma E\cap D\) and \(u(x)=0\) on the set \(\Gamma F\cap D\); 3) \(0\le u(x)\le 1\) inside \(K\).
The closure of \(U(K)\) in the norm \(L_p^{(1)}(K)\) \((p>1)\) will be denoted by \(U_p(K)\). From the well-known theorem of Whitney \((^2)\) on the possibility of extending a sufficiently smooth function given on an arbitrary closed set to the whole space with preservation of smoothness, it follows easily that the set \(U(K)\) is nonempty.
Definition 4. Let \(K\) be an arbitrary conductor. The \(n\)-dimensional \(p\)-conductivity of the conductor \(K\) \((p>1)\) is the number
\[
c_p^{(n)}(K)=a_{p,n}\inf_{u(x)\in U_p(K)}\int_K |\operatorname{grad}u|^p\,dx,
\]
where
\[
a_{p,n}=\omega_n^{-1}\left(\frac{p-1}{|\,n-p\,|}\right)^{p-1}
\quad \text{for } n\ne p,
\qquad
a_{n,n}=\omega_n^{-1};
\]
\(\omega_n\) is the surface area of the unit sphere in the space \(E_n\).
Definition 5. If the set \(\Gamma K\cap \Omega\) (see Definition 2) is closed, then we shall sometimes call \(K\) a condenser. Instead of the term \(p\)-conductivity, in this case we shall speak of the \(p\)-capacity of the condenser. If the sets \(D\) and \(F\) coincide with the entire space \(E_n\), then, naturally, one uses the term \(p\)-capacity of the set \(E\)\(***\).
In proving the theorems of the work \((^1)\) and of the present note, use is made of the direct consequence, formulated below, of the well-known formula
\(*\) The definitions and notation of the note \((^1)\) are somewhat changed here.
\(**\) A subset \(E\) of a set \(D\) is called closed in \(D\) if all points of \(D\) that are limit points for \(E\) belong to \(E\).
\(**\*) On 2-conductivity and 2-capacity see \((^3)\); the definitions in \((^3)\) differ somewhat from those given above.
A. S. Kronrod \((^4)\) for the \(n\)-dimensional variation of a continuously differentiable function.
Lemma 1. Let \(u(x)\) be a function defined and continuously differentiable in \(D\), and let \(\Phi(x)\) be a measurable function in \(D\). Then
\[ \int_{E_c}\Phi(x)|\operatorname{grad}u(x)|\,dx = \int_c^{+\infty}dt\int_{S_t}\Phi(x)\,d\sigma(x), \]
where \(E_c=\{x;\ u(x)\geq c\}\); \(S_c=\{x;\ u(x)=c\}\); \(\int_{S_c}\cdots d\sigma(x)\) is the integral over the set \(S_c\) with respect to \((n-1)\)-dimensional Hausdorff measure.
We give some properties of \(p\)-conductivity.
- The \(n\)-dimensional \(p\)-conductivity of a conductor \(K\) can be defined by the equality
\[ c_p^{(n)}(K) = a_{p,n}\inf_{u(x)\in U(K)} \left\{ \int_0^1 \frac{dt}{ \left(\displaystyle\int_{S_t}|\operatorname{grad}u|^{p-1}\,ds\right)^{\frac{1}{p-1}} } \right\}^{1-p}. \]
- For the \(p\)-capacity of a condenser \(K\) the estimate holds
\[ c_p^{(n)}(K)\geq \begin{cases} v_n^{\frac{p-n}{n}} \left| \operatorname{mes}_n^{-\frac{p-n}{n(p-1)}}(F) - \operatorname{mes}_n^{-\frac{p-n}{n(p-1)}}(E) \right|^{1-p}, & \text{if } n\ne p,\\[6pt] \left(\ln\dfrac{\operatorname{mes}_n F}{\operatorname{mes}_n E}\right)^{1-n}, & \text{if } n=p, \end{cases} \]
where \(v_n\) is the volume of the \(n\)-dimensional ball of unit radius. The equality sign is attained if the condenser is bounded by two concentric spheres. In particular, the capacity of an \(n\)-dimensional ball of radius \(R\) is equal to \(R^{n-p}\) for \(n>p\) and to \(0\) for \(n\leq p\).
- Let \(c_p^{(n)}(K)<\infty\). There exists a unique element \(u^*(x)\in U_p(K)\) such that
\[ c_p^{(n)}(K)=a_{p,n}\int_K|\operatorname{grad}u^*|^p dx. \]
For the element \(u^*(x)\), for almost all \(t\),
\[ \int_{S_t}|\operatorname{grad}u^*|^{p-1}d\sigma=\mathrm{const}. \]
Definition 6. Consider two conductors: \(K=F\setminus E\), \(K'=F'\setminus E'\). We shall say that the conductor \(K'\) is a part of \(K\) \((K'\subset K)\), if \(E\subseteq E'\subset F'\subseteq F\).
We formulate two more properties of \(p\)-conductivity:
-
If \(K'\subset K\), then \(c_p^{(n)}(K)\leq c_p^{(n)}(K')\).
-
For every \(\varepsilon>0\), for an arbitrary conductor \(K\) having finite \(p\)-conductivity, one can construct a conductor \(K'\subset K\) such that
\[ c_p^{(n)}(K')\leq c_p^{(n)}(K)+\varepsilon. \]
\(2^\circ\). The concept of \(p\)-conductivity turns out to be very useful in studying the connection between embedding theorems and sets on which functions are defined.
Definition 7. A set \(D\) belongs to the class \(I_p^{(n)}\) \((p>n)\), if there exists a positive constant \(M<\operatorname{mes}_n D\) such that
\[ \inf c_p^{(n)}(K)=\mathfrak{p}(M)>0, \]
where the infimum is taken over all conductors \(K\) in \(D\) such that \(\operatorname{mes}_n F\leq M\).
It can be shown that Definition 7 is equivalent to the following definition:
Definition 7′. A set \(D\) belongs to the class \(I_p^{(n)}\) if there exists a positive constant \(R\) such that
\[ \inf c_p^{(n)}(K_R)=\mathfrak{P}'(R)>0 . \]
Here the infimum is taken over all conductors \(K_R=F\setminus E\) in \(D\), where \(F=S_R\cap D\), and \(E\) is the center of the \(n\)-dimensional ball \(S_R\) of radius \(R\).
In note \((^1)\) a sufficient condition was formulated for the boundedness and complete continuity of the embedding operator \(\widetilde L_p^{(1)}(D)\cap L(D)\) into \(C(\overline D)\). In terms of the classes \(I_p^{(n)}\) one can formulate a necessary and sufficient condition. Namely, the following theorem is true:
Theorem 1. If a set \(D\) belongs to the class \(I_p^{(n)}\) and there exist constants \(\mathfrak{R}(M)>0\) and \(k>0\) such that for all \(M'\leq M\)
\[ \mathfrak{P}(M')\,M'^k\geq \mathfrak{R}(M), \tag{1} \]
then almost everywhere in \(D\) the inequality
\[ |u|\leq \{k_1\|\operatorname{grad}u\|_{L_p(D)}+k_2\|u\|_{L(D)}\}^{\frac{p}{k+p}}\|u\|_{L(D)}^{\frac{k}{k+p}} \tag{2} \]
holds.
The functions \(u(x)\in L_p^{(1)}(D)\cap L(D)\) \([u(x)\in \widetilde L_p^{(1)}(D)\cap L(D)]\) coincide almost everywhere with functions continuous in \(D\) \([\overline D]\).
Conversely, if for every function \(u(x)\in L_p^{(1)}(D)\cap L(D)\) inequality (2) holds, then the set \(D\) belongs to the class \(I_p^{(n)}\) and (1) holds.
Corollary 1. For boundedness of the embedding operator \(L_p^{(1)}(D)\cap L(D)\) into \(C(D)\), where \(C(D)\) is the space of functions continuous and uniformly bounded in \(D\), it is necessary and sufficient that \(D\in I_p^{(n)}\).
One can give an example of a domain \(D\) \((\operatorname{mes}_n D<\infty)\) such that the embedding operator \(L_p^{(1)}(D)\cap L(D)\) into \(C(D)\) is bounded, but not completely continuous.
Below we formulate a criterion for complete continuity of the embedding operator \(\widetilde L_p^{(1)}(D)\cap L(D)\) into \(C(\overline D)\).
Definition 8. A set \(D\) belongs to the class \(\overline I_p^{(n)}\) \((p>n)\) if \(D\in I_p^{(n)}\) and \(\mathfrak{P}(M)\to\infty\) as \(M\to 0\).
One can define the class \(\overline I_p^{(n)}\) by the condition \(\mathfrak{P}'(R)\to\infty\) as \(R\to 0\).
Theorem 2. Let \(\operatorname{mes}_n D<\infty\). For complete continuity of the embedding operator \(\widetilde L_p^{(1)}(D)\cap L(D)\) into \(C(\overline D)\) it is necessary and sufficient that \(D\in \overline I_p^{(n)}\). If \(D\in I_p^{(n)}\), then the embedding operator \(L_p^{(1)}(D)\cap L(D)\) into \(C(D)\) is completely continuous.
Remark 1. It is easy to show that if at each point \(P\) of the set \(D\) one can construct a cone lying inside \(D\) and given, for a suitable choice of a rectangular coordinate system with origin at the point \(P\), by the inequalities
\[ \left\{\left(\sum_{i=1}^{n-1} x_i^2\right)^{1/2}<\varepsilon x_n^\beta,\; 0<x_n<\rho\right\}\quad(\beta\geq 1), \]
where \(\varepsilon,\rho,\beta\) are fixed, then \(D\in I_p^{(n)}\), and (1) holds for
\[ k=-1+\frac{p}{\beta(n-1)+1}. \]
Definition 9. A set \(D\) belongs to the class \(J_{\nu(t)}^{(n)}\) if there exists a positive constant \(M<\operatorname{mes}_n D\) and a function \(\nu(t)\geq 0\) such that, for almost all \(t\) for which \(\operatorname{mes}_n E_t\leq M\),
\[ \sup \frac{\nu(\operatorname{mes}_n E_t)}{\operatorname{mes}_{n-1} S_t}=\mathfrak{B}(M)<\infty, \]
where the \(\sup\) is taken over all functions \(u(x)\in C^{(1)}(D)\cap L_1^{(1)}(D)\). For \(\nu(t)=t^\alpha\left(\alpha\ge \frac{n-1}{n}\right)\), following remark \((1)\), we shall denote the class \(J_{\nu(t)}^{(n)}\) by \(J_\alpha^{(n)}\).
Let us give an example of a domain in the class \(J_{\nu(t)}^{(n)}\).
Example 1. The \(n\)-dimensional cone
\[ \left\{\left(\sum_{i=1}^{n-1} x_i^2\right)^{1/2}< f(x_n),\ 0<x_n<1\right\}, \]
where \(f(x)\in C^{(1)}([0,1])\), \(f'(x)\) is an increasing function satisfying the condition \(f(0)=0\), belongs to the class \(J_{\nu(t)}^{(n)}\) if the function \(\nu(t)\) satisfies the condition
\[ \nu\left(\alpha_{n-1}\int_0^x f^{\,n-1}(\tau)\,d\tau\right)\le f^{\,n-1}(x). \]
Here \(\alpha_{n-1}\) is the \((n-1)\)-dimensional Lebesgue measure of the \((n-1)\)-dimensional ball of unit radius.
Theorem 3. Let \(D\in J_{\nu(t)}^{(n)}\) and let \(\Phi(t)\) be an \(N\)-function \((5)\) such that
\[ \int_0^M \Psi\left(\frac{1}{\nu(t)}\right)\,dt<\infty, \]
where \(\Psi(t)\) is the function complementary to \(\Phi(t)\), and let \(u(x)\in C^{(1)}(D)\cap L(D)\). Then from the boundedness of the integral
\[ \int_D \Phi(|\operatorname{grad} u|)\,dx \tag{3} \]
it follows that the function \(u(x)\) is bounded. If \(\operatorname{mes} D<\infty\), then the set of functions for which the integral \((3)\) is bounded is compact in \(C(D)\).
Corollary 2. Let \(D\in J_\alpha^{(n)}\) \((\alpha<1)\) and
\[ \int_D |\operatorname{grad} u|^{\frac{1}{1-\alpha}} \left(\ln_{m+1}^+|\operatorname{grad} u|\right)^r \left(\prod_{i=-1}^{m}\ln_i^+|\operatorname{grad} u|\right)^{\frac{\alpha}{1-\alpha}}dx<\infty, \tag{4} \]
where \(m\ge -1\) is an integer; \(r>\frac{\alpha}{1-\alpha}\); \(\ln_i^+t=\ln^+(\ln_{i-1}^+t)\) for \(i\ge 2\); \(\ln_1^+t=\ln t\) for \(t\ge 1\); \(\ln_1^+t=0\) for \(t<1\); \(\ln_0^+t=\ln_{-1}^+t=1\). Then the function \(u(x)\) is bounded in \(D\). If \(\operatorname{mes}_n D<\infty\), then the set of functions for which the integral \((4)\) and \(\|u\|_{L(D)}\) are bounded is compact in \(C(D)\).
For \(r=\frac{\alpha}{1-\alpha}\) Corollary 2 is false. Let us also note that if \(D\in J_1^{(n)}\), then the embedding operator \(L_p^{(1)}(D)\cap L(D)\) into \(C(D)\) can be unbounded for all \(p\).
Leningrad State University
named after A. A. Zhdanov
Received
22 III 1961
REFERENCES
- V. G. Maz’ya, DAN, 133, No. 3 (1960).
- H. Whitney, Trans. Am. Math. Soc., 36, 1 (1934).
- G. Pólya, G. Szegő, Isoperimetric Inequalities in Mathematical Physics, Princeton, 1951.
- A. S. Kronrod, UMN, 5, 1 (35) (1950).
- M. A. Krasnosel’skii, Ya. B. Rutitskii, Convex Functions and Orlicz Spaces, 1958.