BOUNDARY BEHAVIOR OF SOLUTIONS OF A CLASS OF GENUINELY NONLINEAR HYPERBOLIC SYSTEMS
We study the set of boundary singularities of arbitrary classical solutions of genuinely nonlinear $2\times2$ planar hyperbolic systems of the form $D_kR_k=0$, where $D_k$ denotes differentiation in the direction $e^{i\theta_k(R_1,R_2)}$, $k=1,2$, and where the defining functions $\theta_k$ satisfy...
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| Veröffentlicht in: | SIAM journal on mathematical analysis 2008, Vol.40 (4), p.1291-1336 |
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| description | We study the set of boundary singularities of arbitrary classical solutions of genuinely nonlinear $2\times2$ planar hyperbolic systems of the form $D_kR_k=0$, where $D_k$ denotes differentiation in the direction $e^{i\theta_k(R_1,R_2)}$, $k=1,2$, and where the defining functions $\theta_k$ satisfy (i) $\theta_2=\theta_1+\frac{\pi}{2}$, and (ii) $A\leq|\frac{\partial\theta_k(R)}{\partial R_k}| \leq B$, for all $R\in\mathbb{R}^2$, $k=1,2$, for some positive constants $A,B$. We show that for any system of this kind there is a $\tau |
| doi_str_mv | 10.1137/070705507 |
| format | Article |
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We show that for any system of this kind there is a $\tau<1$ such that for any locally Lipschitz solution $R$ in a smoothly bounded domain $G$, the set of points of $\partial G$ at which $R$ fails to have a nontangential limit has Hausdorff dimension at most $\tau$, and, on the other hand, for any such system for which the $\theta_k\in C^\infty(\mathbb{R}^2)$, we construct a $C^\infty$ solution $R$ on a half-plane $\mathbb{H}$ for which the set of points of $\partial\mathbb{H}$ at which $R$ fails to have a nontangential limit has positive Hausdorff dimension. These results are immediately applicable to constant principal strain mappings, which are defined in terms of a system of this kind for which $\theta_1$ is a linear function of $R_1$ and $R_2$.</description><identifier>ISSN: 0036-1410</identifier><identifier>EISSN: 1095-7154</identifier><identifier>DOI: 10.1137/070705507</identifier><language>eng</language><publisher>Philadelphia, PA: Society for Industrial and Applied Mathematics</publisher><subject>Exact sciences and technology ; Global analysis, analysis on manifolds ; Mathematical analysis ; Mathematics ; Partial differential equations ; Sciences and techniques of general use ; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds</subject><ispartof>SIAM journal on mathematical analysis, 2008, Vol.40 (4), p.1291-1336</ispartof><rights>2009 INIST-CNRS</rights><rights>[Copyright] © 2008 Society for Industrial and Applied Mathematics</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c321t-83e09803d37999e6d144ec62b8fb56a0a864474e2e9c137dbf1962228f745ce23</citedby><cites>FETCH-LOGICAL-c321t-83e09803d37999e6d144ec62b8fb56a0a864474e2e9c137dbf1962228f745ce23</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,3185,4024,27923,27924,27925</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=21718622$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>GEVIRTZ, Julian</creatorcontrib><title>BOUNDARY BEHAVIOR OF SOLUTIONS OF A CLASS OF GENUINELY NONLINEAR HYPERBOLIC SYSTEMS</title><title>SIAM journal on mathematical analysis</title><description>We study the set of boundary singularities of arbitrary classical solutions of genuinely nonlinear $2\times2$ planar hyperbolic systems of the form $D_kR_k=0$, where $D_k$ denotes differentiation in the direction $e^{i\theta_k(R_1,R_2)}$, $k=1,2$, and where the defining functions $\theta_k$ satisfy (i) $\theta_2=\theta_1+\frac{\pi}{2}$, and (ii) $A\leq|\frac{\partial\theta_k(R)}{\partial R_k}| \leq B$, for all $R\in\mathbb{R}^2$, $k=1,2$, for some positive constants $A,B$. We show that for any system of this kind there is a $\tau<1$ such that for any locally Lipschitz solution $R$ in a smoothly bounded domain $G$, the set of points of $\partial G$ at which $R$ fails to have a nontangential limit has Hausdorff dimension at most $\tau$, and, on the other hand, for any such system for which the $\theta_k\in C^\infty(\mathbb{R}^2)$, we construct a $C^\infty$ solution $R$ on a half-plane $\mathbb{H}$ for which the set of points of $\partial\mathbb{H}$ at which $R$ fails to have a nontangential limit has positive Hausdorff dimension. These results are immediately applicable to constant principal strain mappings, which are defined in terms of a system of this kind for which $\theta_1$ is a linear function of $R_1$ and $R_2$.</description><subject>Exact sciences and technology</subject><subject>Global analysis, analysis on manifolds</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Partial differential equations</subject><subject>Sciences and techniques of general use</subject><subject>Topology. Manifolds and cell complexes. 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Manifolds and cell complexes. 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Science Collection</collection><collection>ProQuest Central Basic</collection><jtitle>SIAM journal on mathematical analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>GEVIRTZ, Julian</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>BOUNDARY BEHAVIOR OF SOLUTIONS OF A CLASS OF GENUINELY NONLINEAR HYPERBOLIC SYSTEMS</atitle><jtitle>SIAM journal on mathematical analysis</jtitle><date>2008</date><risdate>2008</risdate><volume>40</volume><issue>4</issue><spage>1291</spage><epage>1336</epage><pages>1291-1336</pages><issn>0036-1410</issn><eissn>1095-7154</eissn><abstract>We study the set of boundary singularities of arbitrary classical solutions of genuinely nonlinear $2\times2$ planar hyperbolic systems of the form $D_kR_k=0$, where $D_k$ denotes differentiation in the direction $e^{i\theta_k(R_1,R_2)}$, $k=1,2$, and where the defining functions $\theta_k$ satisfy (i) $\theta_2=\theta_1+\frac{\pi}{2}$, and (ii) $A\leq|\frac{\partial\theta_k(R)}{\partial R_k}| \leq B$, for all $R\in\mathbb{R}^2$, $k=1,2$, for some positive constants $A,B$. We show that for any system of this kind there is a $\tau<1$ such that for any locally Lipschitz solution $R$ in a smoothly bounded domain $G$, the set of points of $\partial G$ at which $R$ fails to have a nontangential limit has Hausdorff dimension at most $\tau$, and, on the other hand, for any such system for which the $\theta_k\in C^\infty(\mathbb{R}^2)$, we construct a $C^\infty$ solution $R$ on a half-plane $\mathbb{H}$ for which the set of points of $\partial\mathbb{H}$ at which $R$ fails to have a nontangential limit has positive Hausdorff dimension. These results are immediately applicable to constant principal strain mappings, which are defined in terms of a system of this kind for which $\theta_1$ is a linear function of $R_1$ and $R_2$.</abstract><cop>Philadelphia, PA</cop><pub>Society for Industrial and Applied Mathematics</pub><doi>10.1137/070705507</doi><tpages>46</tpages><oa>free_for_read</oa></addata></record> |
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| source | LOCUS - SIAM's Online Journal Archive |
| subjects | Exact sciences and technology Global analysis, analysis on manifolds Mathematical analysis Mathematics Partial differential equations Sciences and techniques of general use Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds |
| title | BOUNDARY BEHAVIOR OF SOLUTIONS OF A CLASS OF GENUINELY NONLINEAR HYPERBOLIC SYSTEMS |
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