Optimal Poincare type trace inequalities on the Euclidean ball (Analysis on Shapes of Solutions to Partial Differential Equations)
11
0
0
全文
(2) 105. integrable functions on \partial $\Omega$ with respect to the (n-1) ‐dimensional \mathcal{H}^{n-1} Moreover, there exists a constant C depending on $\Omega$ such that space of .. ,. (1.1). Hausdorff. measure. ,. \displaystyle \inf_{c\in} \Vert\overline{u}-c\Vert_{L^{1}(\partial $\Omega$)}\leq C\Vert Du\Vert( $\Omega$). for every. \in. u. BV( $\Omega$) where \Vert Du\Vert( $\Omega$) u [Ma3, Theorem 9.6.4].. stands for the total variation. ,. Du of. gradient. A property of L^{1} norms ensures that the infimum in on \partial $\Omega$ , given by. (1.1). over. $\Omega$ of distributional. is attained when. c. agrees with. a. median of a. \displaystyle \mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$}\overline{u}=\sup\{t\in \mathbb{R} : \mathcal{H}^{n-1}(\{\overline{u}>t\})>\mathcal{H}^{n-1}(\partial $\Omega$)/2\} (see. [CP1,. e.g.. Lemma. 3.1]) Thus, inequality (1.1). is. equivalent. to. \Vert\overline{u}-\mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$}\overline{u}\Vert_{L^{1}(\partial $\Omega$)}\leq C\Vert Du\Vert( $\Omega$) for every u\in BV( $\Omega$) , with the same constant C. Other normalizing operators than \mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$}\overline{u} are admissible in. General assumptions. on an. inequalities. of the form. (3.2).. operator T such that. BV( $\Omega$)\ni u\mapsto T(u)\in \mathbb{R} are. known for. inequality. an. of the form. (1.2). \Vert\overline{u}-T(u)\Vert_{L^{1}(\partial $\Omega$)}\leq C\Vert Du\Vert( $\Omega$). to hold. for. some. u\in BV( $\Omega$) These assumptions can be derived, for [Zi, Lemma 4.1.3]. classical choice for T(u) is the mean value \mathrm{m}\mathrm{v}\partial $\Omega$(\overline{u}) of. constant C , and for every. .. instance, by specializing. an. abstract result from. Besides the median of \overline{u}. on. \partial $\Omega$ , another. \overline{u}. over. \partial $\Omega$ ,. given by. \displaystyle\mathrm{m}\mathrm{v}\partial$\Omega$(\overline{u})=\frac{1}{\mathcal{H}^{n-1}(\partial$\Omega$)}\int_{\partial$\Omega$}\overline{u}(x)d\mathcal{H}^{n-1}(x). Less conventional admissible operators. T(u). (1.3). T(u)=\mathrm{m}\mathrm{e}\mathrm{d}_{ $\Omega$}(u). .. amount to. ,. where. \displaystyle \mathrm{m}\mathrm{e}\mathrm{d}_{ $\Omega$}(u)=\sup\{t\in \mathbb{R}:|\{u>t\}|>| $\Omega$|/2\}, the median of. u. in the whole of $\Omega$ , and. (1.4). T(u)=\mathrm{m}\mathrm{V} $\Omega$(u). ,. where. \displaystyle\mathrm{m}\mathrm{V}$\Omega$(u)=\frac{1}{|$\Omega$|}\int_{$\Omega$}u(x)dx, the. mean. value of. (1.4). make. both. on. We. \overline{u} and. are. \mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$}\overline{u}. ,. constants. u. in the whole of $\Omega$. Here, |\cdot| denotes Lebesgue. inequality (1.2) nonstandard,. The choices. (1.3). and. quantities depending. on u.. concerned with the. or. measure.. in that its left‐hand side combines. mv \partial $\Omega$. (ũ),. or. problem of. \mathrm{m}\mathrm{e}\mathrm{d}_{ $\Omega$}(u). equal certain geometric. the. optimal. \mathrm{m}\mathrm{v} $\Omega$(u). constant C in. (1.2). when. T(u). is either. For any admissible domain $\Omega$ , these optimal constants of isoperimetric type. In the special case when $\Omega$ is ,. or. ..
(3) 106. Euclidean. an. the. scaling. ball,. an. explicit description of the extremal functions is possible. In fact, due to inequalities, we shall deal, without loss of generality, with. invariance of the relevant. the unit ball \mathrm{B}^{n} , centered at 0 , in \mathbb{R}^{n}.. Interestingly, the Poincaré inequalities in question share the same extremals under the con‐ on \mathrm{m}\mathrm{e}\mathrm{d}_{0 $\Omega$}\overline{u}, \mathrm{m}\mathrm{v}\partial $\Omega$(\overline{u}) and \mathrm{m}\mathrm{v}_{ $\Omega$}(u) but take a different, non‐standard form, for \mathrm{m}\mathrm{e}\mathrm{d}_{ $\Omega$}(u) The geometric characterizations of the sharp constants in the Poincaré inequalities are stated Section 2. Section 3 is devoted to the description of the extremals in the Poincaré inequalities. straint in. .. ,. in \mathrm{B}^{n}.. Let. mention that trace. inequalities in Sobolev type spaces, involving optimal constants, extensively investigated in the literature. Contributions along this line of research include [AFV, AMR, BGP, Bro, \mathrm{B}\mathrm{r}\mathrm{F} Ci2, CFNT2, DDM, Esl, MV1, MV2, Mal, Ma2, Ma3, Na, Ro, \mathrm{W}] Sharp forms of Poincaré type inequalities for Sobolev functions and functions of bounded variation, involving norms of u in the whole of $\Omega$ are the object of [BK, \mathrm{B}\mathrm{o}\mathrm{V}, \mathrm{B}\mathrm{r}\mathrm{V}, Cil, DG, DN, EFKNT, ENT, FNT, GW, Le, NR]. us. have been. ,. .. ,. Geometric constants. 2. Let E be. a. measurable set in \mathbb{R}^{n} The essential .. in \mathbb{R}^{n} of the sets of. boundary \partial^{M}E of E. points of densities. set, and \partial^{M}E\subset\partial E the topological ,. 0 and 1 with respect to E boundary of E.. .. is defined. as. the. complement. \partial^{M}E. One has that. is. a. Borel. The set E is said to be of finite perimeter relative to an open set $\Omega$\subset \mathbb{R}^{n} if D$\chi$_{E} , the distributional derivative of the characteristic function $\chi$_{E} of E , is a vector‐valued Radon measure in $\Omega$ with finite total variation in $\Omega$ The perimeter of E relative to $\Omega$ is defined .. (2.1). P(E; $\Omega$)=\Vert D$\chi$_{E}\Vert( $\Omega$\rangle.. A result from. geometric. \mathcal{H}^{n-1}(\partial^{M}E\cap $\Omega$)<\infty ;. theory. measure. Theorem ,. that E is of finite perimeter in $\Omega$ if and. only. if. 4.5.11].. A domain $\Omega$ in \mathbb{R}^{n} will be called admissible if. \mathcal{H}^{n-1}(\partial $\Omega$)<\infty, \mathcal{H}^{n-1}(\partial $\Omega$\backslash. and. \displaystyle \min\{\mathcal{H}^{n-1}(\partial^{M}E\cap\partial $\Omega$), \mathcal{H}^{n-1}(\partial $\Omega$\backslash \partial^{M}E)\}\leq C\mathcal{H}^{n-1}(\partial^{M}E\cap $\Omega$). (2.3) some. ular,. us. P(E; $\Omega$)=\mathcal{H}^{n-1}(\partial^{M}E\cap $\Omega$). \partial^{M} $\Omega$)=0. for. tells. moreover,. (2.2) [Fe,. as. positive. any. If $\Omega$ is. an. domain is. admissible. \mathcal{H}^{n-1}-\mathrm{a}.\mathrm{e}. x\in\partial $\Omega$. domain,. an. [Zi,. Definition. 5.10.1].. In partic‐. admissible domain.. the. boundary. trace. \overline{u} of. a. function. u\in BV( $\Omega$). is well defined for. as. \displaystyle \overline{u}(x)=\lim_{r\rightar ow 0^{+} \frac{1}{|B_{r}(x)\cap $\Omega$|}\backslash \int_{B_{r}(x)\cap $\Omega$}u(y)dy,. (2.4) where. constant C and every measurable set E\subset $\Omega$. Lipschitz. B_{r}(x). denotes the ball centered at. x,. with radius. r. [Ma3, Corollary 9.6.5].. The. assumption. admissible domain is necessary and sufficient for \overline{u} to belong to L^{1}(\partial $\Omega$) for every function u\in BV( $\Omega$)- see [AG] and [Ma3, Theorem 9.5.2]. Moreover, L^{1}(\partial $\Omega$) cannot be replaced that $\Omega$ be. an. with any smaller Lebesgue space independent of u. Alternate notions of the boundary trace of a function of bounded variation literature. One definition relies upon the notion of upper and lower. can. be found in the. approximate limits of the.
(4) 107. extension of. u. trace in the. sense. by. 0 outside $\Omega$. [Ma3,. of. [Zi,. Definition. Section. 9.5.1].. 5.10.5].. Another possible definition is that of rough. Both of them coincide with \overline{u} , up to subsets of \partial $\Omega$ of. \mathcal{H}^{n-1} ‐measure zero. If $\Omega$ is a Lipschitz domain, and the function u enjoys some additional regularity property, such as membership to the Sobolev space W^{1,1}( $\Omega$) then the trace of u on \partial $\Omega$ defined as the limit of ,. the restrictions to \partial $\Omega$ of. for. \mathcal{H}^{n-1}-\mathrm{a}.\mathrm{e} point .. We denote. on. approximating. sequences of smooth functions. on. St also agrees with \overline{u}. \partial $\Omega$.. assume through this section that $\Omega$ is an admissible by C(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$}) the optimal constant in the inequality. (2.5). domain in \mathbb{R}^{n} , with n\geq 2 Let .. \Vert\overline{u}-\mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$}(\overline{u})\Vert_{L^{1}(\partial $\Omega$)}\leq C(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$})\Vert Du\Vert( $\Omega$). u\in BV( $\Omega$) A pioneering result by Burago and Mazya [Ma3, Theorem 9.5.2] C(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$}) equals the geometric constant K(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$}) of $\Omega$ defined as for. .. tells. us. that. K(\displaystyle\mathrm{ }\mathrm{e}\mathrm{d}_{\partial$\Omega$})=\sup_{E\subset$\Omega$}\frac{\min\{ mathcal{H}^{n-1}(\partial^{M}E\cap\ artial$\Omega$),\mathcal{H}^{n-1}(\partial$\Omega$\backslash\partial^{M}E)\}{\mathcal{H}^{n-1}(\partial^{M}E\cap$\Omega$)}.. (2.6) Here, and tended. us. in similar. over. occurrences. non‐negligible. [Ma3,. Theorem 2.1. in what. follows,. we. tacitly. assume. that the supremum is. ex‐. subsets E of $\Omega$.. Theorem. 9.5.2]. (2.7). Let $\Omega$ be. an. admissible domain in \mathbb{R}^{n} , with n\geq 2. C(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$})=K(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$}). .. Then. .. Equahty holds in (2.5) for some nonconstant function u if and only if the supremum is attained in (2.6) for some set E. In particular, if E\dot{u} an extremal set in (2.6), then the function a$\chi$_{E}+b \dot{u} an extremal function in (2.5) for every a\in \mathbb{R}\backslash \{0\} and b\in \mathbb{R}. More generally, if \{E_{k}\} is an optimizing sequence of sets in (2.6), then the sequence \{u_{k}\} \{a_{k}$\chi$_{E_{k}}+b_{k}\} is an optimizing sequence offunctions in (2.5) for every a_{k}, b_{k}\in \mathbb{R}. =. Let. us. next consider the. (2.8). optimal. constant. C(\mathrm{m}\mathrm{v}\partial $\Omega$). in the. inequality. \Vert\overline{u}-\mathrm{m}\mathrm{v}\partial $\Omega$(\overline{u})\Vert_{L^{1}(\partial $\Omega$)}\leq C(\mathrm{m}\mathrm{v}\partial $\Omega$)\Vert Du\Vert( $\Omega$). u\in BV( $\Omega$) It has K(\mathrm{m}\mathrm{v}\partial $\Omega$) given by. for. .. been shown in. [Ci3]. that. C(\mathrm{m}\mathrm{v}\partial $\Omega$). agrees with another. geometric. constant. ,. (2.9). K(\displayst le\mathrm{ }\mathrm{v}\partial$\Omega$)=\sup_{E\subset$\Omega$}\frac{2\min\{ mathcal{H}^{n-1}(\partial^{M}E\cap\ artial$\Omega$),\mathcal{H}^{n-1}(\partial$\Omega$\backsla h\partial^{M}E)\}{\mathcal{H}^{n-1}(\partial$\Omega$)\mathcal{H}^{n-1}(\partial^{M}E\cap$\Omega$)}.. Theorem 2.2. [Ci3,. (2.10). Theorem. 1.1]. Let $\Omega$ be. an. admissible domain in \mathbb{R}^{n} , with n\geq 2. C(\mathrm{m}\mathrm{v}\partial $\Omega$)=K(\mathrm{m}\mathrm{v}\partial $\Omega$). .. Then. .. Equality holds in (2.8) for some nonconstant function u if and only if the supremum is attained in (2.9) for some set E. In particular, if E\dot{u} an extremal set in (2.9), thenrthe function a$\chi$_{E}+b is an extremal function in (2.8) for every a\in \mathbb{R}\backslash \{0\} and b\in \mathbb{R}. More generally, if \{E_{k}\} is an optimizingI sequence of sets in (2.9), then the sequence \{u_{k}\} \{a_{k}$\chi$_{E_{k}}+b_{k}\} is an optimizing sequence of functions in (2.8) for every a_{k}, b_{k}\in \mathbb{R}. =.
(5) 108. The geometric constants associated with the Poincaré inequalities with normalization de‐ pending on the whole funtion u instead of just its boundary trace \overline{u} are exhibited in [CFNT2]. ,. Specifically,. let. us. denote. (2.11) for $\Omega$. ,. by C(\mathrm{m}\mathrm{v} $\Omega$). the. optimal. constant in the. inequality. \Vert\overline{u}-\mathrm{m}\mathrm{V} $\Omega$(u)\Vert_{L^{1}(\partial $\Omega$)}\leq C(\mathrm{m}\mathrm{v} $\Omega$)\Vert Du\Vert( $\Omega$). u\in BV( $\Omega$). .. Then. C(\mathrm{m}\mathrm{v} $\Omega$). is related to the. isoperimetric. constant. K(\mathrm{m}\mathrm{v} $\Omega$). associated with. by. K(\displaystyle\mathrm{ }\mathrm{v}_{$\Omega$})=\sup_{E\subset$\Omega$}\frac{|E\mathcal{H}^{n-1}(\partial$\Omega$\backslash\partial^{M}E)+|$\Omega$\backslashE|\mathcal{H}^{n-1}(\partial^{M}E\cap\ artial$\Omega$)}{|$\Omega$|\mathcal{H}^{n-1}(\partial^{M}E\cap$\Omega$)}.. (2.12). [CFNT2,. Theorem 2.3. Theorem. 2.1]. (2.13). Let $\Omega$ be. an. admissible domain in \mathbb{R}^{n} , with n\geq 2. C(\mathrm{m}\mathrm{v}_{ $\Omega$})=K(\mathrm{m}\mathrm{v}_{ $\Omega$}). Equality. (2.11) for some. holds in. in. (2.12) for. is. an. nonconstant. .. Then. .. function uif.and only if the. supremum \dot{u} attained. particular, function a$\chi$_{E}+b (2.12), function in (2.11) for every a\in \mathbb{R}\backslash \{0\} and b\in \mathbb{R}. More generally, if \{E_{k}\} is an optimizing sequence of sets in (2.12), then the sequence \{uk\} \{ak$\chi$_{E_{k}}+b_{k}\} \dot{u} an optimizing sequence of functions in (2.11) for every a_{k}, b_{k}\in \mathbb{R}. set E. In. some. if E is. an. extremal set in. then the. extremal. =. We conclude this section in the. (2.14) for. by. a. u\in BV( $\Omega$). The isoperimetric constant which. [ \mathrm{C} $\Gamma$ \mathrm{N}\mathrm{T}2. ,. Theorem. (2.16). into. play. is defined. as. 2.3]. Let $\Omega$ be. an. admissible domain in \mathbb{R}^{n} , with n\geq 2. set E. In. some. extremal in. (2.14) for. generally, if \{E_{k}\}. \{a_{k}$\chi$_{E_{k}}+b_{k}\}. \dot{u}. .. Then. .. (2.14) for some. holds in. (2.15) for an. now comes. C(\mathrm{m}\mathrm{e}\mathrm{d}_{ $\Omega$})=K(\mathrm{m}\mathrm{e}\mathrm{d}_{ $\Omega$}). Equality. More. C(\mathrm{m}\mathrm{e}\mathrm{d}_{ $\Omega$}). K(\displaystle\mathrm{ }\mathrm{e}\mathrm{d}_ $\Omega$})=|E\leq|$\Omega$|/2\sup_{E\subet$\Omega$}\frac{\mathcal{H}^{n-1}(\partil^{M}E\cap\ artil$\Omega$)}{\mathcal{H}^{n-1}(\partil^{M}E\cap$\Omega$)}.. Theorem 2.4. is. constant. \Vert\overline{u}-\mathrm{m}\mathrm{e}\mathrm{d}_{ $\Omega$}(u)\Vert_{L^{1}(\partial $\Omega$)} \leq C(\mathrm{m}\mathrm{e}\mathrm{d}_{ $\Omega$})\Vert Du\Vert( $\Omega$) .. (2.15). in. geometric characterization of the optimal. inequality. an. nonconstant function u if and only if the supremum is attained particular, if E\dot{u} an extremal in (2.15), then the function a$\chi$_{E}+b. every. a\in \mathbb{R}\backslash \{0\}. and b\in \mathbb{R}.. optimizing sequence of sets in (2.15), then the sequence \{uk\} optimizing sequence offunctions in (2.14) for every a_{k}, b_{k}\in \mathbb{R}. \dot{u}. an. =. Remark 2.5 The Poincaré type trace inequalities considered in the present section hold, in particular, with the same constants, for every function u in the Sobolev space W^{1,1}( $\Omega$) In‐ .. deed,. the latter is. agrees with. a. For any such function u , the total variation \Vert Du\Vert( $\Omega$) where \nabla u denotes the weak gradient of u The constants in the relevant. subspace. \Vert\nabla u\Vert_{L^{1}( $\Omega$)}. ,. of. BV( $\Omega$). .. .. inequalities continue to be optimal in W^{1,1}( $\Omega$) since any function u\in BV( $\Omega$) approximated by a sequence of functions u_{k}\in W^{1,1}( $\Omega$) in such a way that Poincaré. ,. \overline{u_{k} =\overline{u}. and. \displaystyle \lim_{k\rightar ow\infty}\Vert\nabla u_{k}\Vert_{L^{1}( $\Omega$)}=\Vert Du\Vert( $\Omega$). can. be. .. The existence of the sequence \{u_{k}\} follows, for instance, from [Gi, Theorem 1.17 and Remark 1,18]. Of course, the last part of the statements of Theorems 2.1−2.4 does not apply when. dealing. with Sobolev. differentiable.. functions,. since characteristic functions of subsets of $\Omega$. are. not. weakly.
(6) 109. Figure. 1. Extremal functions in Poincaré. 3. In the. case. inequalities. on. \mathrm{B}^{n}. when the domain $\Omega$ is the ball \mathrm{B}^{n} , the extremal subsets in geometric functionals can be exhibited. As a consequence of Theorems 2.1−2.4, the extremal. introduced in section 2. functions in the associated Poincaré trace The computation of the. (3.1) for. optimal. inequalities. constant. can. C(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial \mathrm{B}^{n} ). be characterized.. in the Poincaré trace. inequality. \Vert\overline{u}-\mathrm{m}\mathrm{e}\mathrm{d}_{\mathrm{B}^{n} ( $\gamma$ u\Vert_{L^{1}(\partial \mathrm{B}^{n})}\leq C(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial \mathrm{B}^{n} )\Vert Du\Vert(\mathrm{B}^{n}). u\in BV(\mathrm{B}^{n}). In what. goes back to. Theorem 3.1. Burago and Mazya [Ma3, Corollary 9.4.4/3]. the Lebesgue measure of the unit ball. $\omega$_{n}=$\pi$^{n/2}/ $\Gamma$(1+n/2). follows,. ,. [Ma3, Corollary 9.4.4/3]. Let n\geq 2. .. in \mathbb{R}^{n}.. Then. C(\displaystyle\mathrm{ }\mathrm{e}\mathrm{d}_{\partial\mathrm{B}^{n})=\frac{n$\omega$_{n}{2$\omega$_{n-1}. Equality. holds in. (3.1) if u. The best constant. (3.2) for. u. agrees with the characteristic. C(\mathrm{m}\mathrm{v}\partial \mathrm{B}^{n}). in the. function of. a. half‐ball (see Figuoe1).. inequality. \Vert\overline{u}-\mathrm{m}\mathrm{v}\partial \mathrm{B}^{n}(\overline{u})\Vert_{L^{1}(\partial B)} \leq C(\mathrm{m}\mathrm{v}_{\partial \mathrm{B}^{n} )\Vert Du\Vert(B) \in. BV(B). Interestingly,. is. provided by. a. result from. [Ci3],. which is stated in Theorem 3.2 below.. the existence and the form of extremals in. the dimension. n. .. In. particular,. inequality (3.2). need not exist, even for domains with such a simple geometry In what follows, we call spherical segment in B the (non. half‐space.. depend on inequality (2.8) \mathbb{R}^{2}.. turns out to. Theorem 3.2 shows that extremals in the trace as. the disk in. empty). intersection of B with. a.
(7) 110. Figure. Theorem 3.2. [Ci3,. Theorem. 1.2]. (3.3). Let B be. C(\mathrm{m}\mathrm{v}\partial \mathrm{B}^{n})=. ,. holds in. ,. holds in. If n\geq 4 equality If n=3 equality. (3.2) (3.2). 2. ball in. a. \mathbb{R}^{n}, n\geq 2 Then .. \left{\begin{ar y}{l \frac{n$\omega$_{n}2$\omega$_{n-1}&ifn\geq3,\ 2&ifn=2. \end{ar y}\right.. when. u. agrees with the characteristic. when. u. agrees with the characteristic. function of a half‐ball. function of any spherical. segment. 2 equality never holds If n functions of spherical segments =. ,. Remark 3.3 Let. incidentally. us. admissible domains in \mathbb{R}^{n} ,. Indeed,. as. shown in. in. as. (3.2),. whose. far. unless. measure. u. is constant.. converges to 0 is. mention that \mathrm{B}^{n} as. the constants. enjoys. a. Any. sequence. optimizing. in. of characteristic. (3.2).. minimizing property,. C(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial \mathrm{B}^{n} ). and. C(\mathrm{m}\mathrm{v}\partial \mathrm{B}^{n}). are. among all. concerned.. [CFNTI],. (3.4). C(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial $\Omega$})\geq C(\mathrm{m}\mathrm{e}\mathrm{d}_{\partial \mathrm{B}^{n} ). ,. and. (3.5). C(\mathrm{m}\mathrm{v}\partial $\Omega$)\geq C(\mathrm{m}\mathrm{v}\partial \mathrm{B}^{n}). .. Moreover, equality holds in (3.4) if and only if $\Omega$=\mathrm{B}^{n} and, when n\geq 3 equality holds in (3.5) if and only if $\Omega$=\mathrm{B}^{n} On the other hand, if n=2 there also exist domains $\Omega$\neq \mathrm{B}^{2} attaining equality in (3.5). ,. .. Let. us. next focus. on. the estremal functions in Poincaré type inequalities on \mathrm{B}^{n} under mean over the entire \mathrm{B}^{n} In the former case, namely in inequalities of the. value and median constraint. .. form. (3.6). ,. ,. \Vert\overline{u}-\mathrm{m}\mathrm{v}\mathrm{B}^{n}(u)\Vert_{L^{1}(\partial \mathrm{B}^{n})}\leq C(\mathrm{m}\mathrm{v}_{\mathrm{B}^{n} )\Vert Du\Vert(\mathrm{B}^{n}).
(8) 111. Figure. for. u\in BV(\mathrm{B}^{n}). for the. sharp. characteristic functions of half‐balls. holds in. (3.6) if. In contrast with the. Then. .. C(\displaystyle\mathrm{m}\mathrm{v}\mathrm{B}^{n})=\frac{n$\omega$_{n}{2$\omega$_{n-1}. u. agrees with the characteristic. function of. u\in BV(\mathrm{B}^{n}). ,. with. optimal. .. even convex.. half‐ball. in the. inequality. constant. are. C(\mathrm{m}\mathrm{e}\mathrm{d}_{\mathrm{I}\mathrm{B}^{n} ). half‐m6on. ,. are. characteristic functions of. shaped (Figure 2), and hence,. in. a new. kind of. particular, they. are. This is the content of the next theorem.. In the statement, to the set. E_{ $\theta,\ \varphi$}. denotes the set. (3.8). depicted. in. Figure 3,. Theorem 3.5 Let. where the. couple ( $\theta$, $\varphi$) belongs. \mathrm{T}=\{( $\theta$, $\varphi$):0< $\theta$< $\pi$, 0\leq $\varphi$< $\theta$\}.. The isoperimetric nature of the optimal constant in inequality 2.4, helps in accounting for this seemingly striking conclusion.. the. a. \Vert\overline{u}-\mathrm{m}\mathrm{e}\mathrm{d}_{\mathrm{B}^{n} (u)\Vert_{L^{1}(\partial \mathrm{B}^{n})}\leq C(\mathrm{m}\mathrm{e}\mathrm{d}_{\mathrm{B}^{n} )\Vert Du\Vert(\mathrm{B}^{n}). subsets of \mathrm{B}^{n} These subsets not. again extremals. Here, C(\mathrm{m}\mathrm{v}_{\mathrm{B}^{n} ) stands. previous results of this section, the extremals. (3.7) for. are. (3.6).. constant in. Theorem 3.4 Let n\geq 2. Equality. 3. n. half‐moon shaped. \geq 2 set. .. Then. E_{ $\theta,\ \varphi$}. as. equality holds in (3.7) if u Figure 3, where ( $\theta$, $\varphi$) \dot{u}. in. (3.7),. as. described in Theorem. is the characteristic. the. unique solution. function of in the set \mathrm{T}.
(9) 112. (defined by (3.8)). to the. system. \left{bginary}{l \frac$Psi_{n-2}($\varphi)}{$\Ps_n-2}($\thea)}frc{\sin^}$\thea}{sin^ $\varphi}=1-\frac{(n1)$\Psi_{n-2}($\pi)cos$\thea}{2[(n-1)\cos$thea\Psi$_{n-2}(\thea$)-sin^{1}$\thea]}\ frac{os$\varphi}{\sn$varphi}=\fac{os$\thea}{sin$\thea}(1-\frc{n1)$\Psi_{n-2}($\pi){2[(n-1\cos^{2}$\theaPsi$_{n-2}(\thea$)-sin^{1}$\theacos$\thea]}). \end{ary}\ight.. (3.9). Acknowledgements.. This research was partly supported by the research project of MIUR Education, University and Research) Prin 2012, n. 2012\mathrm{T}\mathrm{C}7588 Elliptic and parabolic partial differential equations: geometric aspects, related inequalities, and applica‐ tions and by GNAMPA of the Italian INdAM (National Institute of High Mathematics).. (Italian Ministry. of. ,. References. [AFV] A.Alvino,. A.Ferone &. R.Volpicelli, Sharp Hardy inequalities. remainder term, Nonlinear Anal. 75. [AMR] F.Andreu, J.M.Mazón, from. [AG]. W^{1,1}( $\Omega$). G.Anzellotti &. (1978), [BK]. & J.D.Rossi, The best constant for the Sobolev trace Nonlinear Anal. 59 (2004), 1125‐1145. ,. L^{1}(\partial $\Omega$). M.Giaquinta,. with trace. embedding. Funzioni BVe tracce, Rend. Sem. Mat. Univ. Padova 60. M.Belloni &. B.Kawohl, A symmetry problem Diff. Eq. 156 (1999), 211‐218.. J.. [BGP] E.Berchio, ary. half‐space. 1‐21.. equality,. [BoV]. into. (2012),. in the. 5466‐5472.. F.Gazzola &. conditions,. related to. Wirtingers. D.Pierotti, Gelfand type elliptic problems under Steklov bound‐. Ann. Inst. Henri Poincaré Anal. Non Linéaire. V.Bouchez & J.Van. and Poincarés in‐. Schaftingen,. 27. (2010),. Extremal functions in Poincaré‐Sobolev. 315‐335.. inequalities. for. functions of bounded variation, in Nonlinear Elliptic Partial Differential Equations, Amer. Math.. [BrF]. Mathematics. 540, 2011,. 47‐58.. L.Brasco &. Wulff. [BrV]. Soc., Contemporary. G.Franzina, An anisotropic eigenvalue problem inequalities, Nonhnear Diff. Equat. Appl. (NoDEA) ,. of Stekloff type and 20. (2013),. weighted. 1795‐1830.. H.Brezis & J.Van. Schaftingen, Circulation integrals and critical Sobolev spaces: problems sharp constants, Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math. 79, Amer. Math. Soc. Providence, RI, 2008, of. 33‐47.. [Bro]. F.. Brock, An isoperimetric inequality. Math. Mech. 81. [BM] Yu.D.Burago. &. (2001),. for. eigenvalues. of the Stekloff. problem,. Z.. Angew.. 69−71.. V.G.Mazya, Some questions of potential theory and function theory for non‐regular boundaries, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 3 (1967), 1‐152 (Russian); English translation: Seminars in Mathemat‐ ics, V.A. Steklov Mathematical Institute, Leningrad, Consultants Bureau, New York, 3. domains with. (1969). 1‐68..
(10) 113. [Cil] A.Cianchi, Math.. A. sharp form of. Phys. (ZAMP). 40. Poincaré type (1989), 558‐569.. [Ci2] A.Cianchi, Moser‐Trudinger [Ci3] A.Cianchi, Royal. sharp trace inequality Edinburgh 142A (2012),. [CFNTI] A.Cianchi, V.Ferone, J. Reine. Angew.. 217. [CP1]. A.Cianchi &. (2008),. Angew.. Z.. 2005‐2044.. ball, Proc.. 1179‐1191.. C.Trombetti,. 725. (2017),. Balls minimize trace constants in. C.Nitsch & C.Trombetti, Poincaré normalization, preprint.. L.Pick, Sobolev embeddings. BV,. 41‐61.. [\mathrm{C} $\Gamma$ \mathrm{N}\mathrm{T}2] A.Cianchi, V.Ferone, with nonstandard. spheres,. for functions of bounded variation in the. C.Nitsch &. (Crelle J.). Math.. balls and. on. inequalities, Adv. Math.. trace. A. Soc.. inequalites. into spaces of. inequalities. trace. in. BV(\mathrm{B}^{n}). Campanato, Morrey, and. Hölder. type, J. Math. Anal. Appl. 282 (2003), 128‐150.. [DDM] J.Davila, L.Dupaigne problem, Comm.. [DG]. F.Della Pietra &. Pure. & M.Montenegro, The extremal solution Appl. Analysis 7 (2008) 795‐817.. N.Gavitone, Symmetrization for. Neumann. (2012),. related questions, Advanced Nonlinear Studies 12. of. a. boundary. reaction. anisotropic problems and. 219‐235.. [DN] A.V.Demyanov. & A.I.Nazarov, On the existence of an extremal function in Sobolev em‐ bedding theorems with a limit exponent, Algebra i Analiz 17 (2005), 105‐140 (Russian); English translation: St. Petersburgh Math. J. 17 (2006), 773‐796.. [Esl] J.F.Escobar, Sharp (1988), 687‐698.. [Es2] J.F.Escobar, 165. (1999),. [ENT] L.Esposito, domains,. constant in. An isoperimetric. a. Sobolev trace. inequality. and the first Steklov. C.Nitsch &. J. Convex. sharp. Indiana Univ. Math. J. 37. eigenvalue,. J. Funct. Anal.. 101‐116.. C.Trombetti, Best constants Analysis 20 (2013), 253‐264.. [EFKNT] L.Esposito, V.Ferone, B.Kawohl, and. inequality,. Poincaré‐Sobolev. C.Nitsch &. inequalities,. in Poincaré. inequalities. for. convex. C.Trombetti, The longest shortest fence. Arch. Rational. Mech. Anal. 206. (2012),. 821‐. 851.. [Fe]. H.Federer, Geometric. [FNT] V.Ferone, for. [GW]. convex. P. Girão & in. a. ball,. measure. theory. Springer, Berlin,. 1969.. C.Nitsch & C.Trombetti, A remark on optimal weighted Poincaré inequalities domains, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appt. 23 (2012), 467‐475.. T.Weth, The shape of extremal functions for Poincaré‐Sobolev‐type inequalities. J. Funct. Anal. 237. (2006),. 194‐233.. [Gi]. E.Giusti, Minimal surfaces. [Ka]. B.Kawohl, Rearrangements and convexity of level 1150, Springer‐Verlag, Berlin, 1985.. [Le]. M.Leckband, A rearrangement based proof for the existence of extremal functions for the Sobolev‐Poincaré inequality on B^{n} J. Math. Anal. Appl. 363 (2010), 690‐696.. and functions of bounded variation. ,. sets in PDE. Birkhäuser, Basel,. 1984,. Lecture Notes in Math..
(11) 114. [MV1] F.Maggi & C.Villani, (2005), 83−121.. Balls have the worst best Sobolev constants, J. Geom. Anal. 15. [MV2] F.Maggi. & C.Villani, Balls have the worst best Sobolev inequalities II. Variants and extensions, Calc. Var. Partial Differential Equations 31 (2008), 47‐74.. [Mal] V.G.Mazya,. Classes of. Nauk. SSSR 133. (1960),. regions and imbedding theorems for function spaces, Dokl. Akad. 527‐530 (Russian); English translation: Soviet Math. Dokl. 1. (1960),. 882‐885.. [Ma2] V.G.Mazya,. Classes of sets and. imbedding. theorems for function spaces, Dissertation. MGU, Moscow, 1962 (Russian).. [Ma3] V.G.Mazya,. Sobolev spaces with. Springer, Heidelberg,. [NR]. B.Nazaret, 65. [Ro]. to. elliptic partial. differential. equations. A.I.Nazarov & zero mean. [Na]. applications. 2011.. (2006),. J.D.Rossi,. S.I.Repin, Exact constants in Poincar type inequalities for functions boundary traces, Math. Methods Appl. Sci. 38 (2015), 3195‐3207. Best constant in Sobolev trace. inequalities. on. the. half‐space,. with. Nonlinear Anal.. 1977‐1985.. First variations of the best Sobolev trace constant with respect to the (2008), 140‐145.. domain,. Canad. Math. Bull. 51. [W]. R.. Weinstock, Inequalities. (1954),. [Zi]. for. a. classical. eigenvalue problem,. J. Rational Mech. Anal. 3. 745‐753.. W.P.Ziemer, Weakly differentiable functions. Springer‐Verlag,. New. York,. 1989..
(12)
関連したドキュメント
One can distinguish several types of cut elimination proofs for higher order logics/arith- metic: (i) syntactic proofs by ordinal assignment (e.g. Gentzen’s consistency proof for
Equivalent conditions are obtained for weak convergence of iterates of positive contrac- tions in the L 1 -spaces for general von Neumann algebra and general JBW algebras, as well
Wall theorems give local lower bounds for the p-measure of the boundary of a domain in the euclidean n -space.. We improve earlier results by replacing the euclidean metric by the
In particular, in view of the results of Guillemin [16] [17], this means that on Toeplitz operators T Q of order ≤ −n, the Dixmier trace Tr ω T Q coincides with the residual trace
Secondly, we establish some existence- uniqueness theorems and present sufficient conditions ensuring the H 0 -stability of mild solutions for a class of parabolic stochastic
[2] Agmon S., Douglis A., Nirenberg L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm..
The commutative case is treated in chapter I, where we recall the notions of a privileged exponent of a polynomial or a power series with respect to a convenient ordering,
Then it follows immediately from a suitable version of “Hensel’s Lemma” [cf., e.g., the argument of [4], Lemma 2.1] that S may be obtained, as the notation suggests, as the m A
