משפט רליך-קונדרשוב

באנליזה פונקציונלית, משפט רליך-קונדרשוב הוא משפט לגבי שיכון קומפקטי (כלומר, שיכון רציף שהוא גם אופרטור קומפקטי) בין שני מרחבי סובולב. המשפט קרוי על שם המתמטיקאי הגרמני-אוסטרי פרנץ רליך והמתמטיקאי הרוסי ולדימיר קונדרשוב.

ניסוח המשפט

תהי $\Omega \subset \mathbb {R} ^{n}$ קבוצה פתוחה, חסומה וליפשיצית ויהי $1\leq p<n$ .

נגדיר $p^{*}={\frac {np}{n-p}}$ .

אזי מרחב הסובולב $W^{1,p}(\Omega )$ ניתן לשיכון רציף במרחב ה-Lp $L^{P^{*}}(\Omega )$ ולשיכון קומפקטי במרחב $L^{q}(\Omega )$ לכל $1\leq q\leq p^{*}$ .

כלומר, $W^{1,p}(\Omega )\hookrightarrow L^{p^{*}}(\Omega )$ וגם $1\leq q<p^{*}\implies W^{1,p}(\Omega )\subset \subset L^{q}(\Omega )$ .

תוצאות

היות ששיכון הוא קומפקטי אם ורק אם אופרטור השיכון (הזהות) הוא אופרטור קומפקטי, נובע ממשפט רליך-קונדרשוב שלכל סדרה חסומה במידה שווה במרחב $W^{1,p}(\Omega )$ קיימת תת-סדרה המתכנסת במרחב $L^{q}(\Omega )$ . המסקנה הזאת ידועה כמשפט הבחירה של רליך-קונדרשוב.

משפט רליך-קונדרכוב שימושי להוכחת אי-שוויון פואנקרה^[1] לפיו לכל $u\in W^{1,p}(\Omega )$ (כאשר $\Omega$ עומד בתנאי משפט רליך-קונדרכוב) מקיים:

$\|u-u_{\Omega }\|_{L^{p}(\Omega )}\leq C\|\nabla u\|_{L^{p}(\Omega )}$

כאשר הקבוע C תלוי רק בערך p ובתכונות הגאומטריות של $\Omega$ וכן

u_{\Omega }{\overset {\mbox{def}}{=}}{\frac {1}{\mathrm {vol} (\Omega )}}\int _{\Omega }u(x)\,\mathrm {d} x

הוא הערך הממוצע של u בתחום $\Omega$ .

הערות שוליים

^ Evans, Lawrence C. (2010). "§5.8.1". Differential Equations, Partial (2nd ed.). p. 290. ISBN 0-8218-4974-3.

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