Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

Handle URI:
http://hdl.handle.net/10754/623172
Title:
Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary
Authors:
Chen, Huyuan; Alhomedan, Suad; Hajaiej, Hichem; Markowich, Peter A. ( 0000-0002-3704-1821 )
Abstract:
The purpose of this paper is to study the weak solutions of the fractional elliptic problem(Formula presented.) where (Formula presented.), (Formula presented.) or (Formula presented.), (Formula presented.) with (Formula presented.) is the fractional Laplacian defined in the principle value sense, (Formula presented.) is a bounded (Formula presented.) open set in (Formula presented.) with (Formula presented.), (Formula presented.) is a bounded Radon measure supported in (Formula presented.) and (Formula presented.) is defined in the distribution sense, i.e.(Formula presented.) here (Formula presented.) denotes the unit inward normal vector at (Formula presented.). In this paper, we prove that (0.1) with (Formula presented.) admits a unique weak solution when g is a continuous nondecreasing function satisfying(Formula presented.) Our interest then is to analyse the properties of weak solution when (Formula presented.) with (Formula presented.), including the asymptotic behaviour near (Formula presented.) and the limit of weak solutions as (Formula presented.). Furthermore, we show the optimality of the critical value (Formula presented.) in a certain sense, by proving the non-existence of weak solutions when (Formula presented.). The final part of this article is devoted to the study of existence for positive weak solutions to (0.1) when (Formula presented.) and (Formula presented.) is a bounded nonnegative Radon measure supported in (Formula presented.). We employ the Schauder’s fixed point theorem to obtain positive solution under the hypothesis that g is a continuous function satisfying(Formula presented.)-pagination
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Chen H, Alhomedan S, Hajaiej H, Markowich P (2017) Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary. Complex Variables and Elliptic Equations: 1–43. Available: http://dx.doi.org/10.1080/17476933.2016.1278441.
Publisher:
Informa UK Limited
Journal:
Complex Variables and Elliptic Equations
Issue Date:
6-Feb-2017
DOI:
10.1080/17476933.2016.1278441
Type:
Article
ISSN:
1747-6933; 1747-6941
Sponsors:
Suad Alhemedan extends his appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group NO (RGP- RG-1438-047). H. Chen is supported by NNSF of China [grant number 11401270], [grant number 11661045]; Jiangxi Provincial Natural Science Foundation [grant number 20161ACB20007].
Additional Links:
http://www.tandfonline.com/doi/full/10.1080/17476933.2016.1278441
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorChen, Huyuanen
dc.contributor.authorAlhomedan, Suaden
dc.contributor.authorHajaiej, Hichemen
dc.contributor.authorMarkowich, Peter A.en
dc.date.accessioned2017-04-13T11:50:59Z-
dc.date.available2017-04-13T11:50:59Z-
dc.date.issued2017-02-06en
dc.identifier.citationChen H, Alhomedan S, Hajaiej H, Markowich P (2017) Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary. Complex Variables and Elliptic Equations: 1–43. Available: http://dx.doi.org/10.1080/17476933.2016.1278441.en
dc.identifier.issn1747-6933en
dc.identifier.issn1747-6941en
dc.identifier.doi10.1080/17476933.2016.1278441en
dc.identifier.urihttp://hdl.handle.net/10754/623172-
dc.description.abstractThe purpose of this paper is to study the weak solutions of the fractional elliptic problem(Formula presented.) where (Formula presented.), (Formula presented.) or (Formula presented.), (Formula presented.) with (Formula presented.) is the fractional Laplacian defined in the principle value sense, (Formula presented.) is a bounded (Formula presented.) open set in (Formula presented.) with (Formula presented.), (Formula presented.) is a bounded Radon measure supported in (Formula presented.) and (Formula presented.) is defined in the distribution sense, i.e.(Formula presented.) here (Formula presented.) denotes the unit inward normal vector at (Formula presented.). In this paper, we prove that (0.1) with (Formula presented.) admits a unique weak solution when g is a continuous nondecreasing function satisfying(Formula presented.) Our interest then is to analyse the properties of weak solution when (Formula presented.) with (Formula presented.), including the asymptotic behaviour near (Formula presented.) and the limit of weak solutions as (Formula presented.). Furthermore, we show the optimality of the critical value (Formula presented.) in a certain sense, by proving the non-existence of weak solutions when (Formula presented.). The final part of this article is devoted to the study of existence for positive weak solutions to (0.1) when (Formula presented.) and (Formula presented.) is a bounded nonnegative Radon measure supported in (Formula presented.). We employ the Schauder’s fixed point theorem to obtain positive solution under the hypothesis that g is a continuous function satisfying(Formula presented.)-paginationen
dc.description.sponsorshipSuad Alhemedan extends his appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group NO (RGP- RG-1438-047). H. Chen is supported by NNSF of China [grant number 11401270], [grant number 11661045]; Jiangxi Provincial Natural Science Foundation [grant number 20161ACB20007].en
dc.publisherInforma UK Limiteden
dc.relation.urlhttp://www.tandfonline.com/doi/full/10.1080/17476933.2016.1278441en
dc.subjectDirac massen
dc.subjectFractional Laplacianen
dc.subjectGreen kernelen
dc.subjectRadon measureen
dc.subjectSchauder’s fixed point theoremen
dc.titleComplete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundaryen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalComplex Variables and Elliptic Equationsen
dc.contributor.institutionDepartment of Mathematics, Jiangxi Normal University, Nanchang, P.R. Chinaen
dc.contributor.institutionDepartment of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabiaen
dc.contributor.institutionNew York University Shanghai, Shanghai, Chinaen
kaust.authorMarkowich, Peter A.en
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