Kato's inequality when $\Delta_{p}u$ is a measure and related topics (Analysis on Shapes of Solutions to Partial Differential Equations)

全文

(1)128. 数理解析研究所講究録第2046巻 2017年 128-139. Katos. inequality. when. $\Delta$_{p}u. is. a measure. and related topics. 堀内. 利郎 (Toshio HORRIUCHI). 劉 Ibaraki. 暁静 (Xiaojing LIU). University, Mito, Ibaraki, Japan. 平成29年1月16日. 目次 1. Introduction. 2. 2. Main Aim. 3. 3. Decomposition of Radon. 4. Definition of admissible class. 5. 5. Some results. 5. on. the. measures. admissibility. 5. 6. Counter‐example. due to J.Serrin. 6. 7. Main results and. Applications. 6. 8. Existence of admissible solution. 10. 9. Problems. 10. 1.

(2) 129. Introduction. 1 $\Omega$. :. a. bounded domain of. \mathrm{R}^{N} (N\geq 1). .. $\Delta$ u=\mathrm{d}\mathrm{i}\mathrm{v}(\nabla u) , \nabla u=(\partial u/\partial x_{1}, \partial u/\partial x_{2}, \ldots, \partial u/\partial x_{N}) Convex type. 1.1. inequality. (The classical convex type Katos inequality) Let L_{1\mathrm{o}\mathrm{c} ^{1}( $\Omega$),-then $\Delta$|u| and $\Delta$ u^{+} are Radon measures and we have. Lemma 1. where. $\Delta$|u|\geq \mathrm{s}\mathrm{g}\mathrm{n}(u) $\Delta$ u. in. D'( $\Omega$). $\Delta$ u^{+}\geq$\chi$_{[u\geq 0]} $\Delta$ u. in. D'( $\Omega$). \mathrm{s}\mathrm{g}\mathrm{n}(s)=1 if s>0,. Remark 1.1. 1.. If we. -1. if s<0. assume. and. zero. in addition that. $\Delta$|u|=\mathrm{s}\mathrm{g}\mathrm{n}(u) $\Delta$ u The. inequality (1). is. consequence. a. 2. Similar. when. ;. $\Delta$|u| \geq \mathrm{s}\mathrm{g}\mathrm{n}(u) $\Delta$ u. of the fact. that. |u|. in. $\Delta$ u is. L_{1\mathrm{o}\mathrm{c} ^{1}( $\Omega$). s.t.. $\Delta$ u. \in. (1) (2). ,. at. s=0u^{+}=\displaystyle \max[u, 0].. u. is continuous in $\Omega$ , then. in. D'([u\neq 0. we. have. (3). D'( $\Omega$) on. the set. [u=0].. replaced by elhptic operator M(x, \partial_{x}) :. a_{j,k}(x)\in C^{1}. Concave type. Definition 1. \in. inequalities hold. ,. and. for. some. ( aj,k(x) 亀 ku ). ). C>0. \displaystyle \sum_{j.k=1}a_{j,k}(x)$\xi$_{j}$\xi$_{k}\geq C| $\xi$|^{2} 1.2. u. ,. takes its minimum. M(x,\displaystyle\partial_{x})u=\sum_{j.k=1}^{N}\partial_{x_{j} where. .. ,. for. any. $\xi$\in \mathrm{R}^{N}. inequality. (Thuncation): T_{k}(s). :. Given k>0 ,. we. denote. by T_{k}:\mathrm{R}\rightarrow \mathrm{R}. a. truncation. function. T_{k}(\mathcal{S}):=. \left{\begin{ar y}{l k&ifs\geqk,\ s&if-k<sk,\ -k&ifs\leq-k. \end{ar y}\right. 2. (4).

(3) 130. Since. T_{k}|_{\mathrm{R}_{+}. is concave,. Lemma 2 Assume that any. k\geq 0. we. u\in. have the. following. L_{1\mathrm{o}\mathrm{c} ^{1}( $\Omega$). $\Delta$ u\in. ,. lemma:. L_{1\mathrm{o}\mathrm{c} ^{1}( $\Omega$). $\chi$ s(x). is. a. characteristic. Moreover, \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} $\Delta$ u. can. butional derivatives of. be. in. function of S\subset. replaced by $\Delta$_{p}u. u\in L_{1\mathrm{o}\mathrm{c} ^{1}( $\Omega$). Let 1.. 2.. (Classical) For u\in K_{p}( $\Omega$). 1<p\leq\infty. .. D'( $\Omega$). .. in $\Omega$. .. Then, for. (5). ,. under additional. assumptions. on. distri‐. by. we. ,. have. (Convex type):. $\Delta$_{p}|u| \geq \mathrm{s}\mathrm{g}\mathrm{n}(u)$\Delta$_{p}u. in. D'( $\Omega$). $\Delta$_{p}u^{+}\geq x[u\geq 0\mathrm{J}^{$\Delta$_{p}u}. in. D'( $\Omega$). (Concave type): If u\geq 0. ,. the. we. Here. K_{p}( $\Omega$). is. (6). ,. (7). .. have. $\Delta$_{p}T_{k}(u)\leq$\chi$_{[0\leq u\leq k]}$\Delta$_{p}u. in. D'( $\Omega$). |\nabla u|^{p\rightar ow 2}|\partial_{j,k}^{2}u|\in L_{loc}^{1}( $\Omega$) where. p^{*}=\displaystyle \max[(p-1), 1].. 2. Main Aim Consider. a. (8). .. given by. K_{p}( $\Omega$)=\{u\in L_{loc}^{1}( $\Omega$):\partial_{j}u, \partial_{j,k}^{2}u\in L_{loc}^{p^{*} ( $\Omega$). class of second order. improved Katos inequalities. for j, k=1 2, ,. ,. N },. elhptic operators \mathcal{A} including $\Delta$_{p}. when Au is. a. Radon. 3. (9). ,. A: $\Omega$\times R^{N}\mapsto R^{N} satisfies the following assumptions for. \mathcal{C}1, c_{2} and c3:. and estabhsh. measure.. Au=\mathrm{d}\mathrm{i}\mathrm{v}A(x, \nabla u) where. a.e. R.. $\Delta$_{p}u=\mathrm{d}\mathrm{i}\mathrm{v}(|\nabla u|^{p-2}\nabla u) 1. \geq 0. .. is defined. Here, p‐Laplace operator. Example. u. have. we. $\Delta$(T_{k}(u))\leq$\chi$_{[0\leq u\leq k]} $\Delta$ u where. and. some. positive numbers.

(4) 131. 1. the function. x\mapsto A(x, $\xi$). is bounded measurable for. 2. the function. $\xi$\mapsto A(x, $\xi$). is continuous for. a.e.. \forall_{ $\xi$}\in R^{N},. x\in $\Omega$,. 3.. \foral _{ $\xi$}, $\eta$\in R^{N}. |A(x, $\xi$)-A(x, $\eta$)| \leq c_{2}(| $\xi$|+| $\eta$|)^{p-2}| $\xi$- $\eta$|,. ,. ae.. x\in $\Omega$,. 4.. (A(x, $\xi$)-A(x, $\eta$))\cdot( $\xi$- $\eta$)\geq c_{3}(| $\xi$|+| $\eta$|)^{p-2}| $\xi$- $\eta$|^{2}, \foral _{ $\xi$}, $\eta$\in R^{N}. ,. a.e.. x\in $\Omega$,. 5.. A(x, $\lambda \xi$)= $\lambda$| $\lambda$|^{p-2}A(x, $\xi$) Remark 2.1. 1. It. follows from. the. A(x, $\xi$)\cdot $\xi$\geq c_{1}| $\xi$|^{p} 2. For. some. for all. ,. $\lambda$\in R, $\lambda$\neq 0.. assumption 4 that. for all. $\xi$\in R^{N}. have. we. and. x\in $\Omega$.. a.e.. C>0. \displayst le\sum_{j,k=1}^{N}|\frac{\parti lA_{j} \parti l$\xi$_{k}(x,$\xi$)|. \leq C| $\xi$|^{p-2},. \forall_{ $\xi$}\in R^{N}\backslash \{0\}. a e .. ,. .. x\in $\Omega$. (10). ,. Then A satisfies the assumptions 3 and 4.. Example. 1. In the. 2. case. of. $\Delta$_{p},\cdot A=A( $\xi$)=| $\xi$|^{p-2} $\xi$. and A. ,. satisfies the. estimate. (10). 2. Assume that aj, k \in. L^{\infty}( $\Omega$). ,. aj) k =a_{k,j} for j, k. l,. =. 2,. .. .. .. ,. N and. \{a_{j,k}\} satisfies. the uniformly elliptic estimate:. \displaystyle\sum_{j,k=1}^{N}a_{j,k}$\xi$_{j}$\xi$_{k}\geqC|$\xi$|^{2} Bu=\displaystyle\sum_{j,k=1}^{N}\frac{\partial}{\partialx_{j} (a_{j,k}(x)|\nablau|^{p-2}\frac{\partialu}{\partialxk}) for. If p with. is. sufficiently. A_{j} (. Definition 2. $\mu$\in M( $\Omega$). x). $\xi$ ). close to. the space. \Leftrightarrow For every open set. \infty. assume. (11). .. the finiteness. 1\sim 5. .. of Radon measure): $\omega$\subset\subset $\Omega$, \exists_{C_{ $\omega$}}>0. \foral _{ $\varphi$\in C_{0}^{\infty}( $\omega$)}. We do not. $\xi$\in R^{N}.. 2, then the operator \mathcal{B} satisfies the assumptions. =\displaystyle \sum_{k=1}^{N}(a_{j,k}(x)| $\xi$|^{p-2}$\xi$_{k}). ( M( $\Omega$) :. any. s.t.. |\displaystyle \int_{ $\Omega$} $\varphi$ d $\mu$| \leq C_{ $\omega$}| $\varphi$| _{L}\infty. of the total measure | $\mu$|( $\Omega$)<\infty but. for each $\omega$\subset\subseteq $\Omega$. 4. assume. ,. for. | $\mu$|( $\omega$)<.

(5) 132. Decomposition of Radon. 3. $\mu$\in M( $\Omega$) (see e.g. [7, 10]). For any on. $\Omega$. Total. ,. be. measures. uniquely decomposed. $\mu$. can. :. $\mu$=$\mu$_{d}+$\mu$_{c} where. as a sum. of two Radon. measures. ,. \left{bginary}{l $\mu_{d}(A)=0\mathr{f}\mathr{o}\mathr{}\mathr{}\mathr{n}\mathr{y}\mathr{B}\mathr{o}\mathr{}\mathr{e}\mathr{l}\mathr{s}\mathr{e}\mathr{}A\subet$Omga\ thrm{s}.\athrm{}C_p(A,$\Omega)=0,\ |$mu_{\athrmc}|($\OmegabckslahF)=0\mathr{f}\mathr{o}\mathr{}\mathr{s}\mathr{o}\mathr{}\mathr{e}\mathr{B}\mathr{o}\mathr{}\mathr{e}\mathr{l}\mathr{s}\mathr{e}\mathr{}F\subet$Omga\ thrm{s}.\athrm{}C_p('F,$\Omega)=0. \end{ary}\ight. | $\mu$|=$\mu$^{+}+$\mu$^{-}. measure:. Definition 3. (Ap‐capacity. For each compact set. K\subset $\Omega$,. C_{p}(K, $\Omega$)=\displaystyle \inf { ($\mu$_{d})^{+}=($\mu$^{+})_{d}. Note that. relative to $\Omega$ ). \displayst le\int_{$\Omega$}|\nabla$\varphi$|^{p}:. $\varphi$\in C_{0}^{\infty}( $\Omega$) and. ,. ($\mu$_{c})^{+}=($\mu$^{+})_{c} by. $\varphi$\geq 1. in. some. nbd. of. K }.. the definition.. Definition of admissible class. 4. (Admissible p^{*}=\displaystyle \max(1,p-1). Definition 4 Let. class in. W_{1\mathrm{o}\mathrm{c} ^{1,p^{*} ( $\Omega$) ). .. u\in W_{1\mathrm{o}\mathrm{c} ^{1,p^{*} ( $\Omega$) is said to be admissible sequence \{u_{n}\}_{n=1}^{\infty} \subset W_{1\mathrm{o}\mathrm{c} ^{1,p}( $\Omega$)\cap L^{\infty}( $\Omega$) s.t 1. u_{n}\rightarrow u in $\Omega$, u_{n}\rightarrow u in W_{1\mathrm{o}\mathrm{c} ^{1,p^{*} ( $\Omega$) A. function. iff Au\in M( $\Omega$) and. there exists. a. :. a.e.. 2.. as n\rightarrow\infty.. \mathcal{A}u_{n}\in L_{1\mathrm{o}\mathrm{c} ^{1}( $\Omega$) (n=1,2, \cdots). and. \displaystyle \sup_{n}|Au_{n}|( $\omega$) =\displaystyle \sup_{n}\int_{ $\omega$}|\mathcal{A}u_{n}| Some results. 5. 1. If u\in are. 2.. W_{1\mathrm{o}\mathrm{c} ^{1,p^{*} ( $\Omega$). every $\omega$\subset\subset $\Omega$. (12). .. admissibility. is admissible \Rightarrow. u^{+}. =\displaystyle \max[u, 0],. u^{-} =\displaystyle \max. [ -u 0 ]) ). T_{k}(u). admissible.. T_{k}(u) dent. \in W_{lo\mathrm{c} ^{1,p}( $\Omega$). on. 3. When. for. \foral _{k}. >0. .. Moreover, given. $\omega$. \subset\subset$\omega$'. \subset\subset. $\Omega$, \exists c>. 0. u\mathrm{s}.\mathrm{t}. \left\{begin{ar y}{l \int_{$\omega$}|\nabl T_{k}(u)|^{2}\leqk(\int_{$\omega$},| \Delta$u|+C\int_{$\omega$},|u),\mathrm{i}\mathrm{f}p=2,\ \int_{$\omega$}|\nabl T_{k}(u)|^{p}\leqCk(\int_{$\omega$},|\Delta$_{p}u|+\int_{$\omega$},|\nabl u|^{p-1})\mathrm{i}\mathrm{f}p\neq2, \end{ar y}\right.. p=2 and \mathcal{A}= $\Delta$,. u\in W_{1\mathrm{o}\mathrm{c} ^{1,1}( $\Omega$) 4.. the. on. for. <\infty. u\in W_{0}^{1,p}( $\Omega$). ,. ,. $\Delta$ u\in M( $\Omega$\vec{\underline{)},}\mathrm{u}. is admissible.. \mathcal{A}u\in M( $\Omega$)\Rightarrow \mathrm{u}. is admissible.. 5. indepen‐.

(6) 133. due to J.Serrin. Counter‐example. 6. Let $\Omega$^{\mathrm{Y} \mathrm{b}\mathrm{e}. a. unit ball. B_{1}=\{x\in R^{N} : |x| <1\}. ,. and set. a_{i,j}=$\delta$_{i,j}+(a-1)\displaystyle \frac{x_{i}x_{j} {r^{2} , (r=|x|) Then. we. have. \displaystyle\mathcal{B}u=\sum_{j,k=1}^{N}\frac{\partial}{\partialx_{j}(a_{j,k}(x)\frac{\partialU}{\partialx_{k})=0. a. pathological. U(x)=x_{1}r^{- $\alpha$}. (13). ,. (14). .. weak solution of the form. $\alpha$=\displaystyle \frac{N}{2}+\sqrt{(\frac{N}{2}-1)^{2}+\frac{N-1}{a}. where.. ). (15). .. If a>1 \Rightarrow N-1< $\alpha$<N. 1 Assume that a> 1. Proposition U is not. admissible,. and. \mathcal{B}(U^{+}). Main results and. 7. In the rest of this note,. U\in W_{1\mathrm{o}\mathrm{c} ^{1,1}(B_{1}). Then. .. is not. a. Radon. and \mathcal{B}U=0 in. D'(B_{1}). .. But. measure.. Applications. we assume. for the sake of. simplicity. \mathcal{A}=$\Delta$_{p}. 7.1. Improved Concave type inequality. Theorem 1. [15, 16]. Assume that. u\in W_{1\mathrm{o}\mathrm{c} ^{1,p^{*} ( $\Omega$). and. u. is addmissible.. \Downar ow\Downar ow\Downar ow If u\geq 0. a.e. .. in $\Omega$ , then. $\Delta$_{\mathrm{p} (T_{k}(u). is. a. Radon. measure. for. every k>0. .. Moreover,. have. we. $\Delta$_{p}(T_{k}(u))\leq($\Delta$_{p}u)^{+} 7.2. Application. Theorem 2 a.e. .. 1.. and. u. [15] Let. u=\~{u}. $\Omega$ be. is admissible.. There exists a.e.. a. Strong. to. a. Maximum. (16). .. Principle. bounded domain of \mathrm{R}^{N}. .. u\in W_{1\mathrm{o}\mathrm{c} ^{1,p^{*} ( $\Omega$). Assume that. ,. u\geq 0. Then. quasicontinuous function. in $\Omega$.. 6. (w.r.t. C_{p}). ũ. :. $\Omega$. \mapsto. \mathrm{R} such that.

(7) 134. 2. Assume that. -$\Delta$_{p}u\geq 0 If ũ. =. 0. Remark 7. 1. on some. -$\Delta$_{p}u. in $\Omega$. K\subset $\Omega$ with be. can. in the. C_{p}(K, $\Omega$)>0. sense. ,. then u=0. replaced by-$\Delta$_{p}u+au^{q}. ,. (17). of measures.. where. a.e. .. in $\Omega$.. 0\leq a\in L_{loc}^{1}( $\Omega$). and q\geq. p-1.. Example. 0\}\cap B_{1/2}. =. x_{1}/|x|^{ $\alpha$}. is not admissible in $\Omega$. =. B_{1}. .. Moreover U. =. 0. on. \{x_{1}. =. which has positive p ‐capacity.. A. 7.3. 3 U. quick. sketch of the. Since. proof. $\Delta$_{p}u\leq 0. of Theorem 2. in $\Omega$. in the. measure. sense,. \Downar ow. ($\Delta$_{p}u)_{d}^{+}=0 \Downar ow Since. T_{k}(u). \in W_{1\mathrm{o}\mathrm{c} ^{1,p}( $\Omega$). ,. $\Delta$_{p}(T_{k}(u)) \in M( $\Omega$). $\Delta$_{p}(T_{k}(u))\leq($\Delta$_{p}u)_{d}^{+}=0. in. for any. \mathcal{D}'( $\Omega$). ,. k>0, \forall k>0.. \Downar ow Now. we can assume. that. u\in L^{\infty}( $\Omega$). \Downar ow As. a. test. function, using. $\varphi$_{0}^{p}/(u+ $\delta$)^{p-1}. with $\varphi$_{0}=1. on. $\omega$,. \displaystyle \int_{ $\omega$}|\nabla\log(\frac{u}{ $\delta$}+1)|^{p}\leq C\int_{ $\Omega$}($\varphi$_{0}^{p}+|\nabla$\varphi$_{0}|^{p}). .. \Downar ow Let E\subset $\Omega$ with. C_{p}(E, $\Omega$)>0. s.t. \overline{u}=0. on. E\subset $\omega$\subset\subset $\Omega$.. By the Poincares inequality. \displaystyle \int_{ $\omega$}|\log(\frac{u\prime}{ $\delta$}+1)|^{p}\leq C\int$\varphi$_{0}^{p}+|\nabla$\varphi$_{0}|^{p} \foral $\delta$>0. \Downar ow We conclude that. u=0. a.e.. in $\Omega$. 口. .. 7.

(8) 135. Convex type Katos. 7.4. [15_{f} 16]. Theorem 3. W_{1\mathrm{o}\mathrm{c} ^{1,p^{*} ( $\Omega$). and. Let $\Phi$ be. inequality a. C^{1}. is addmissible. Then. u. convex we. s.t 0 \leq $\Phi$' <. function. 1 Assume the. same. Assume. .. u. \in. have. $\Delta$_{p} $\Phi$(u)\geq$\Phi$'(u)^{p-1}($\Delta$_{p}u)_{d}-| $\Phi$'| _{L(\mathrm{R})}\infty($\Delta$_{p}u)_{\overline{c} Corollary. \infty. D'( $\Omega$). in. (18). .. in Theorem 3.. assumptions. Then it holds that. $\Delta$_{p}(u^{+})\geq$\chi$_{[u\geq 0]}($\Delta$_{p}u)_{d}-($\Delta$_{p}u)_{\overline{c}. in. $\Delta$_{p}|u| \geq \mathrm{s}\mathrm{g}\mathrm{n}(u)($\Delta$_{p}u)_{d}-|$\Delta$_{p}u|_{c} where. \mathrm{s}\mathrm{g}\mathrm{n}(t)=1 for t>0, \mathrm{s}\mathrm{g}\mathrm{n}(t). Example 1.. u. 4 Let. satisfies. ball B_{1}. .. If. 2—. for. $\Delta$_{p}u= $\alpha$| $\alpha$|^{p-2}cN $\delta$, ,. $\delta$. :. a. cN. and. $\Delta$_{p}(u^{+})\geq$\chi$_{[\mathrm{u}\geq 0]}($\Delta$_{p}u)_{d}-($\Delta$_{p}u)_{\overline{\mathrm{c} } 1/N. \leq p \leq. N , then. $\alpha$. (20). ,. =0.. u. the. :. =. surface. area. of the. unit. is addimible.. inD'( $\Omega$). C_{p}(\{0\}, $\Omega$). \leq 0,. sgn(0). (19). ,. and 0\in $\Omega$.. Dirac mass,. |\nabla u|\in L_{lo\mathrm{c} ^{1}( $\Omega$). D'( $\Omega$). in. t<0 , and. u=|x|^{ $\alpha$} for $\alpha$=(p-N)/(p-1). If p>2-1/N then. Recall 2.. =-1. D'( $\Omega$). 0. (19). .. ($\Delta$_{p}(u^{+}). is. concentrated). ($\Delta$_{p}(u^{+}))_{c}=($\Delta$_{p}u)_{c}=-($\Delta$_{p}u)_{\overline{c}}=\mathrm{a}|\mathrm{a}|^{p-2}cN $\delta$\leq 0. If p>N_{f}. then. ($\Delta$_{p}u)_{\overline{\mathrm{c} }=0. $\alpha$>0,. and. C_{p}(\{0\}, $\Omega$)>0 ($\Delta$_{p}(u^{+}). is. diffuse) ($\Delta$_{p}(u^{+}))_{\mathrm{c}}=($\Delta$_{p}u)_{c}=. $\Delta$_{p}(u^{+})=$\chi$_{[u\geq 0]}($\Delta$_{p}u)_{d}= $\alpha$| $\alpha$|^{p-2}cN $\delta$\geq 0.. Consequently. $\Delta$_{\mathrm{p}}(u^{+})=$\chi$_{[u\geq 0]}($\Delta$_{p}u)_{d}-($\Delta$_{p}u)_{L} inD'( $\Omega$) Inverse maximum. 7.5. Theorem 4 and. u. [15, 16] (Inverse. is admissible.. Then. we. principle maximum. principle ) Assume. 2 Assume. u\in. W_{1\mathrm{o}\mathrm{c} ^{1,p^{*} ( $\Omega$),u\geq 0. have. (-$\Delta$_{p}u)_{c}\geq 0 Corollary. .. u\in W_{1\mathrm{o}\mathrm{c} ^{1,p^{*} ( $\Omega$). and. u. on. $\Omega$. (21). .. is admissible.. (-$\Delta$_{p}(u^{+}))_{c}=(-$\Delta$_{p}u)_{c}^{+}. 8. on. Then $\Omega$. .. we. have. (22).

(9) 136. A. 7.6. quick. sketch of the. proof. of Theorem 4:. Recall:. T_{k}(u)\in W_{1\mathrm{o}\mathrm{c} ^{1,p}( $\Omega$) $\Delta$_{p}(T_{k}(u))\in M( $\Omega$). for \forall k>0.. ,. Moreover. we. have. $\Delta$_{p}(T_{k}(u))\leq($\Delta$_{p}u)^{+} Set. $\Delta$_{p}u= $\mu$\in M( $\Omega$). For. some. compact. As. D'( $\Omega$\backslash K). in. $\mu$_{c}|_{K}=$\mu$_{K}\leq 0. Then. |$\mu$_{\mathrm{c} |( $\Omega$\backslash K)=0, C_{p}(K, $\Omega$)=0.. $\mu$=$\Delta$_{p}u\leq$\chi$_{ $\Omega$\backslash K}$\mu$^{+}. k\rightarrow\infty,. .. .. set K , s.t.. $\Delta$_{p}T_{k}(u) \leq$\mu$^{+}. Then. D^{l}( $\Omega$). in. in. D'( $\Omega$). .. D'( $\Omega$). in. .. .. \Downar ow \mathrm{i}\mathrm{n}. $\mu$_{c}\leq 0. D'( $\Omega$). (23). .. 口. Application. 7.7. Theorem 5. of IMP. [17]Suppose. that. u. is admissible.. Then supp $\mu$_{c}^{\pm}. \subset. \{x : u = \pm\infty\} for. $\mu$=$\Delta$_{p}u. Remark 7.2 From this. if u\in W_{1\mathrm{o}\mathrm{c} ^{1,p^{*} ( $\Omega$) then. 7.8. u. is also. A. quick. Suppose. that. is a. an. fact, admissible solution. (local). sketch of u. of of-$\Delta$_{p}u= $\mu$\in M( $\Omega$). renormalized solution. proof. of-$\Delta$_{p}u= $\mu$.. of Theorem 5. is admissible.. \Downar ow. $\Delta$_{p}u=$\Delta$_{p}(T_{k}u)+$\Delta$_{p}(u-k)^{+}-$\Delta$_{p}(u+k)^{-} -$\Delta$_{p} $\mu$=$\mu$_{d}+$\mu$_{\mathrm{c} ^{+}-$\mu$_{\overline{c} \Downar ow. $\Delta$_{p}(u-k)^{+} $\Delta$_{p}(u+k)^{-} ). 9. \leq 0. (IMP). ,.

(10) 137. \Downar ow Note that. $\Delta$_{p}(T_{k}u). is diffuse and k is. an. number.. arbitrary \Downar ow. supp. $\mu$_{c}^{\pm}\subset\{x:u=\pm\infty\} 口. Existence of admissible solution. 8. TheoreM 6. [17]Assumethat $\mu$\in M( $\Omega$). | $\mu$|( $\Omega$). and. \left{\begin{ar y}{l -$\Delta$_{p}u=$\mu$,\ u=0, \end{ar y}\right. has. The. an. W_{0}^{1,p^{*} ( $\Omega$). admissible solution in. proof rehes. Lemma 3 Let. the. on. following. in. <\infty. .. Then. $\Omega$,. on. (24). $\Omega$.. .. lemma.. \{$\mu$_{n}\} satisfy \displaystyle \sup_{n}|$\mu$_{n}|( $\Omega$)<\infty. and. \{u_{n}\}. be admissible. Assume that. \left{\begin{ar y}{l \tex{一}$\Delta$_{p}u_{n}=$\mu$_{n},&\tex{伽}$\Omega$,\ u_{n}=0,&on$\Omega$. \end{ar y}\right. holds. for. Then. n=1 ,. 2,. up to. a. ,. .. .. (25). ... subsequence, u_{n}\rightarrow\exists_{u}\in. (24) for \exists_{ $\mu$}.. W_{0}^{1,p^{*} ( $\Omega$). s.t.. u. is. admissible. and. satisfy. Problems. 9 1.. 2.. (Nonlinear version of Good measure problem) Let g(s) be continuous, nonnegative and nondecreasing on [0, \infty ). When does next equation have an admissible solution? -$\Delta$_{p}u+g(u)= $\mu$, u|_{\partial $\Omega$}=0 (Nonlinear If u,. version of. $\Delta$_{p}u\in L^{1}. Ex: Even if. ,. then. boundary. $\Delta$_{p}u^{+}. p=2 there ,. is. is a. a. Katos. inequality). finite measure?. u\in H^{1}( $\Omega$). 10. ,. s.t.. $\Delta$ u=0 , but. \displaystyle \int_{ $\Omega$}| $\Delta$ u^{+}|=\infty. the.

(11) 138. 参考文献 [1]. A.. Ancona,. Une. propriété dinvariance des ensembles absorbants. bation dun opérateur. [2]. elliptique,. (1979),. Comm. PDE 4,. par. pertur‐. 321‐337.. P.. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J.L. Vazquez, An L^{1}- theory of existence and uniqueness of solutions of nonlinear elliptic. equations, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. série,. [3]. P.. tome. 22, No.2, (1995),. 4e. 241‐273.. Bénilan, H. Brezis, Nonlinear problems related. to the. Thomas‐Fermi equa‐. tion, J. Evol. Equ. 3, (2004), 673‐770.. [4]. L.. Boccardo, T. Gallouët, Nonlinear elliptic and parabohc equations involving. measure. [5]. L.. deta, J. FUnct. Anal. vol. 87, 1989,. Boccardo,. T.. Gallouët,. lutions for nonlinear section \mathrm{C} , tome 13. [6]. H.. H.. H.. (2003),. 16. Brezis,. M.. [9]. Lorenzo \mathrm{D}. on. the strong maximum. ples. for. [10]. L.. Ambrosio,. (2012),. no.. 9‐. T.. Aspects. T.. T.. a. measure,. C. R. Acad. Sci.. measures re‐. of Nonlinear Dispersive Equations. (J.. Bour‐. Annals of Mathematics Studies, 163 NJ. 2007,. Mitidieri, A priori and. p. 55‐110.. estimates and reduction. princi‐. Adv. Differential. Equa‐. applications,. 10, 93520131000.. (N.S.) 10, (2004),. in. elliptic. 341‐358.. p ‐capacity, J. Math. Soc.. Japan, vol. 43,. No.. 3,. 605‐617.. Horiuchi, Some remarks. (2001), [13]. Enzo. Horiuchi, On the relative. (1991), [12]. Differential. Dupaigne and A. Ponce, Singularities of positive supersolutions. PDEs, Selecta Math.. [11]. is. Marcus, A. Ponce, Nonlinear elhptic equations with. quasilinear elliptic problems. tions 17. principle,. 1‐12.. University Press, Princeton, . so‐. de II.H.P.. p. 539‐551.. gain, C. Kenig, and S. Klainerman, eds Princeton. data,. Annales. (2004),599-604.. in Mathematical. visited,. uniqueness of entropy. measure. Brezis, A. Ponce, Katos inequality when $\Delta$ u. Paris, Ser. I 338. [8]. (1996),. Brezis, A. Ponce, Remarks. Integral Equations. [7]. 5. Existence and. equations with. elhptic. no.. Orsina,. L.. 149‐169. on. Katos. inequality,. J. of. Inequal. & Appl., vol. 6,. 29‐36.. Horiuchi, Katos Inequalities for Degenerate Quasilinear Elliptic Operators,. Kyungpook. Mathematical Journal 2008 Vol.. 11. 48, No. 1, 15—24.

(12) 139. [14] [15]. T.. Kato) Schrödinger operators with singular potentials.Israel J. Math.13. (1972),. 135‐148.. X.. T.. Liu,. Laplacian,. [16]. X.. Horiuchi, Remarks. Liu, Toshio Horiuchi, Remarks. Mathematical Journal of Ibaraki. [17]. X.. F.. Kato. operators and its. general. p‐. is. a. measure,. Volume 48, 2017, P. 45‐61.. . inequalities for measure‐valued. \mathrm{s}. application,. of Dirichlet. in. preparation.. problems for quasilinear elliptic. data, Hiroshima Math. J., 38. measure. involving. (November 2016). inequality when $\Delta$_{p}u. \mathrm{s}. University. Maeda, Renormalized solutions. equations with. [19]. on. Liu, Toshio Horiuchi, Improved Kato. quasilineqar elliptic. [18]. the strong maximum principle. on. Hiroshima Mathematical Journal Volume 46, No.3. (2008),. 51‐93.. Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic. G.D.. equations with general. measure. data, Ann. Scuola Norm. Sup. Pisa, 28) (1999),. 741‐808457‐468.. [20]. E.. Stredulinsky, Weighted Inequalities. ential. [21]. J. L.. Equations,. \mathrm{M}. $\Gamma$. .. Vázquez, A strong. maximum. principle. for. [math.AP]. 13 Feb. Bidaut‐Véron. gradient. Partial Differ‐. 1074, 1984,. some. 96‐139.. quasilinear elliptic. equa‐. 191‐202.. Bidaut‐Viron, M. Garcia‐Huidobro, L. Víron, Remarks. linear equations with. [23]. Degenerate Elliptic. Lecture Notes in Mathematics vol.. tions, App. Math. Optim 12, (1984),. [22]. and. terms. and. measure. quasi‐ data, \mathrm{a}r\mathrm{X}\mathrm{i}\mathrm{v}:1211.6542 on some. (2013).. M.F., Nguyen Quoc, /\mathrm{H} and L. Véron Quasilinear Emden‐Fowler. equations with absorption. .. terms and. 12. measure. data, preprint.

(13)