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Shahgholian, Henrikorcid.org/0000-0002-1316-7913

Open this publication in new window or tab >>A free boundary problem with log term singularity### de Queiroz, Olivaine S.

### Shahgholian, Henrik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_0_j_idt183_some",{id:"formSmash:j_idt179:0:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_0_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_0_j_idt183_otherAuthors",{id:"formSmash:j_idt179:0:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_0_j_idt183_otherAuthors",multiple:true}); 2017 (English)In: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 19, no 3, p. 351-369Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Free boundary, regularity theory, logarithmic singularity, porosity
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-220303 (URN)10.4171/IFB/385 (DOI)000416727600002 ()2-s2.0-85031497391 (Scopus ID)
#####

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#####

##### Funder

Swedish Research Council
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

We study a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional (sic)(v) = integral(Omega) (vertical bar del v vertical bar(2)/2 - v(+)(log v - 1))dx -> min which should be minimized in some natural admissible class of non-negative functions. Here, v(+) = max{0, v}. The Euler-Lagrange equation associated with (sic) is -Delta u = chi({u>0}) log u, which becomes singular along the free boundary partial derivative{u > O}. Therefore, the regularity results do not follow from classical methods. Besides, the logarithmic forcing term does not have scaling properties, which are very important in the study of free boundary theory. Despite these difficulties, we obtain optimal regularity of a minimizer and show that, close to every free boundary point, they exhibit a super-characteristic growth like r(2)vertical bar log r vertical bar. This estimate is crucial in the study of analytic and geometric properties of the free boundary.

QC 20171221

Available from: 2017-12-21 Created: 2017-12-21 Last updated: 2017-12-21Bibliographically approvedOpen this publication in new window or tab >>An elliptic free boundary arising from the jump of conductivity### Kim, S.

### Lee, K. -A

### Shahgholian, Henrik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_1_j_idt183_some",{id:"formSmash:j_idt179:1:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_1_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_1_j_idt183_otherAuthors",{id:"formSmash:j_idt179:1:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_1_j_idt183_otherAuthors",multiple:true}); 2017 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 161, p. 1-29Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier Ltd, 2017
##### Keywords

Conductivity jump, Free boundary problem, Quasilinear elliptic equation, Mathematical techniques, Nonlinear analysis, Free boundary, Free-boundary problems, Lipschitz regularity, Matrix-valued, Monotonicity, Quasi-linear elliptic, Quasilinear elliptic equations, Linear equations
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-216189 (URN)10.1016/j.na.2017.05.010 (DOI)000407182400001 ()2-s2.0-85020887578 (Scopus ID)
#####

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#####

##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

In this paper we consider a quasilinear elliptic PDE, div(A(x,u)∇u)=0, where the underlying physical problem gives rise to a jump for the conductivity A(x,u), across a level surface for u. Our analysis concerns Lipschitz regularity for the solution u, and the regularity of the level surfaces, where A(x,u) has a jump and the solution u does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrix-valued conductivity is presented.

QC 20171220

Available from: 2017-12-20 Created: 2017-12-20 Last updated: 2017-12-20Bibliographically approvedOpen this publication in new window or tab >>Boundary behaviour for a singular perturbation problem### Karakhanyan, A. L.

### Shahgholian, Henrik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_2_j_idt183_some",{id:"formSmash:j_idt179:2:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_2_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_2_j_idt183_otherAuthors",{id:"formSmash:j_idt179:2:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_2_j_idt183_otherAuthors",multiple:true}); 2016 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 138, p. 176-188Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2016
##### Keywords

Free boundary problem, Regularity, Contact points
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-187141 (URN)10.1016/j.na.2015.12.024 (DOI)000374009200010 ()2-s2.0-84955297833 (Scopus ID)
#####

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_2_j_idt183_j_idt366",{id:"formSmash:j_idt179:2:j_idt183:j_idt366",widgetVar:"widget_formSmash_j_idt179_2_j_idt183_j_idt366",multiple:true});
#####

##### Funder

Swedish Research Council
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

In this paper we study the boundary behaviour of the family of solutions (uε) to singular perturbation problem δuε=βε(uε),(divides)uε(divides)≤1 in B1+=(xn>0)∩((divides)x(divides)<1), where a smooth boundary data f is prescribed on the flat portion of ∂B1+. Here βε((dot operator))=1εβ((dot operator)ε),β∈C0∞(0,1),β≥0,∫01β(t)=M>0 is an approximation of identity. If ∇f(z)=0 whenever f(z)=0 then the level sets ∂(uε>0) approach the fixed boundary in tangential fashion with uniform speed. The methods we employ here use delicate analysis of local solutions, along with elaborated version of the so-called monotonicity formulas and classification of global profiles.

QC 20160517

Available from: 2016-05-17 Created: 2016-05-17 Last updated: 2017-11-30Bibliographically approvedOpen this publication in new window or tab >>L2-estimates for singular oscillatory integral operators### Aleksanyan, Hayk

### Shahgholian, Henrik

### Sjölin, Per

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_3_j_idt183_some",{id:"formSmash:j_idt179:3:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_3_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_3_j_idt183_otherAuthors",{id:"formSmash:j_idt179:3:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_3_j_idt183_otherAuthors",multiple:true}); 2016 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 441, no 2, p. 529-548Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2016
##### Keywords

Helmholtz equation, Maximal operator, Oscillating surface, Singular integral
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-187381 (URN)10.1016/j.jmaa.2016.04.031 (DOI)000375635300002 ()2-s2.0-84963859665 (Scopus ID)
#####

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_3_j_idt183_j_idt366",{id:"formSmash:j_idt179:3:j_idt183:j_idt366",widgetVar:"widget_formSmash_j_idt179_3_j_idt183_j_idt366",multiple:true});
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). The University of Edinburgh.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of L2L2 type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis to the eigenvalue problem associated with the Helmholtz equation in R3.

QC 20160527

Available from: 2016-05-27 Created: 2016-05-23 Last updated: 2017-06-20Bibliographically approvedOpen this publication in new window or tab >>REGULARITY ISSUES FOR SEMILINEAR PDE-S (A NARRATIVE APPROACH)### Shah gholian, Henrik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_4_j_idt183_some",{id:"formSmash:j_idt179:4:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_4_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_4_j_idt183_otherAuthors",{id:"formSmash:j_idt179:4:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_4_j_idt183_otherAuthors",multiple:true}); 2016 (English)In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 27, no 3, p. 577-587Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Mathematical Society (AMS), 2016
##### Keywords

Pointwise regularity, Laplace equation, divergence type equations, free boundary problems
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:kth:diva-186599 (URN)10.1090/spmj/1405 (DOI)000373930300015 ()2-s2.0-84963541486 (Scopus ID)
#####

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_4_j_idt183_j_idt360",{id:"formSmash:j_idt179:4:j_idt183:j_idt360",widgetVar:"widget_formSmash_j_idt179_4_j_idt183_j_idt360",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_4_j_idt183_j_idt366",{id:"formSmash:j_idt179:4:j_idt183:j_idt366",widgetVar:"widget_formSmash_j_idt179_4_j_idt183_j_idt366",multiple:true});
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Occasionally, solutions of semilinear equations have better (local) regularity properties than the linear ones if the equation is independent of space (and time) variables. The simplest example, treated by the current author, was that the solutions of Delta u = f(u), with the mere assumption that f ' >= -C, have bounded second derivatives. In this paper, some aspects of semilinear problems are discussed, with the hope to provoke a study of this type of problems from an optimal regularity point of view. It is noteworthy that the above result has so far been undisclosed for linear second order operators, with Holder coefficients. Also, the regularity of level sets of solutions as well as related quasilinear problems are discussed. Several seemingly plausible open problems that might be worthwhile are proposed.

QC 20160531

Available from: 2016-05-31 Created: 2016-05-13 Last updated: 2017-11-30Bibliographically approvedOpen this publication in new window or tab >>A general class of free boundary problems for fully nonlinear parabolic equations### Figalli, Alessio

### Shahgholian, Henrik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_5_j_idt183_some",{id:"formSmash:j_idt179:5:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_5_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_5_j_idt183_otherAuthors",{id:"formSmash:j_idt179:5:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_5_j_idt183_otherAuthors",multiple:true}); 2015 (English)In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 194, no 4, p. 1123-1134Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Free boundaries, Regularity, Parabolic fully nonlinear
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-173446 (URN)10.1007/s10231-014-0413-7 (DOI)000359804400009 ()2-s2.0-84931568511 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

In this paper, we consider the fully nonlinear parabolic free boundary problem { F(D(2)u) - partial derivative(t)u = 1 a.e. in Q(1) boolean AND Omega vertical bar D(2)u vertical bar + vertical bar partial derivative(t)u vertical bar <= K a.e. in Q(1)\Omega, where K > 0 is a positive constant, and Omega is an (unknown) open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W-x(2,) (n) boolean AND W-t(1,) (n) solutions are locally C-x(1,) (1) boolean AND C-t(0,) (1) inside Q(1). A key starting point for this result is a new BMO-type estimate, which extends to the parabolic setting the main result in Caffarelli and Huang (Duke Math J 118(1): 1-17, 2003). Once optimal regularity for u is obtained, we also show regularity for the free boundary partial derivative Omega boolean AND Q(1) under the extra condition that Omega superset of{u not equal 0}, and a uniform thickness assumption on the coincidence set {u = 0}.

QC 20150915

Available from: 2015-09-15 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approvedOpen this publication in new window or tab >>An overview of unconstrained free boundary problems### Figalli, Alessio

### Shahgholian, Henrik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_6_j_idt183_some",{id:"formSmash:j_idt179:6:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_6_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_6_j_idt183_otherAuthors",{id:"formSmash:j_idt179:6:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_6_j_idt183_otherAuthors",multiple:true}); 2015 (English)In: Philosophical Transactions. Series A: Mathematical, physical, and engineering science, ISSN 1364-503X, E-ISSN 1471-2962, Vol. 373, no 2050, article id 20140281Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

free boundary problems, obstacle-type, optimal regularity, matching data
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-173962 (URN)10.1098/rsta.2014.0281 (DOI)000360825300008 ()2-s2.0-84939155428 (Scopus ID)
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##### Funder

Swedish Research Council
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

In this paper, we present a survey concerning unconstrained free boundary problems of type F-1(D(2)u, del u, u, x) = 0 in B-1 boolean AND Omega, F-2(D(2)u, del u, u, x) = 0 in B-1 \ Omega, u is an element of S(B-1), where B-1 is the unit ball, Omega is an unknown open set, F-1 and F-2 are elliptic operators (admitting regular solutions), and S is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to the optimal regularity of solutions to the above matching problems, and list several open problems in this direction.

QC 20151006

Available from: 2015-10-06 Created: 2015-09-24 Last updated: 2017-12-01Bibliographically approvedOpen this publication in new window or tab >>Analysis of a free boundary at contact points with Lipschitz data### Karakhanyan, A. L.

### Shahgholian, Henrik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_7_j_idt183_some",{id:"formSmash:j_idt179:7:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_7_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_7_j_idt183_otherAuthors",{id:"formSmash:j_idt179:7:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_7_j_idt183_otherAuthors",multiple:true}); 2015 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 367, no 7, p. 5141-5175Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Contact points, Free boundary problem, Regularity
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-167776 (URN)10.1090/S0002-9947-2015-06187-X (DOI)000357045700022 ()2-s2.0-84927613989 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

In this paper we consider a minimization problem for the functional J(u) = ∫ B+<inf>1</inf> |∇u|2 + λ2 <inf>+</inf>χ<inf>{u>0}</inf> + λ2 <inf>−</inf>χ<inf>{u≤0}</inf> in the upper half ball B+ <inf>1</inf> ⊂ ℝn, n ≥ 2, subject to a Lipschitz continuous Dirichlet data on ∂B+ <inf>1</inf>. More precisely we assume that 0 ∈ ∂{u > 0} and the derivative of the boundary data has a jump discontinuity. If 0 ∈ ∂({u > 0} ∩ B+ <inf>1</inf>), then (for n = 2 or n ≥ 3 and the one-phase case) we prove, among other things, that the free boundary ∂{u > 0} approaches the origin along one of the two possible planes given by γx<inf>1</inf> = ±x<inf>2</inf>, where γ is an explicit constant given by the boundary data and λ± the constants seen in the definition of J(u). Moreover the speed of the approach to γx<inf>1</inf> = x<inf>2</inf> is uniform.

QC 20150526

Available from: 2015-05-26 Created: 2015-05-22 Last updated: 2017-12-04Bibliographically approvedOpen this publication in new window or tab >>Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L-p Estimates### Aleksanyan, Hayk

### Shahgholian, Henrik

### Sjölin, Per

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_8_j_idt183_some",{id:"formSmash:j_idt179:8:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_8_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_8_j_idt183_otherAuthors",{id:"formSmash:j_idt179:8:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_8_j_idt183_otherAuthors",multiple:true}); 2015 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 215, no 1, p. 65-87Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2015
##### Keywords

Elliptic-Systems, Green
##### National Category

Other Mathematics
##### Identifiers

urn:nbn:se:kth:diva-159614 (URN)10.1007/s00205-014-0774-5 (DOI)000347403500002 ()
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##### Funder

Swedish Research Council
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). The University of Edinburgh.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

Let u(epsilon) be a solution to the system div(A(epsilon)(x)del u(epsilon)(x)) = 0 in D, u(epsilon)(x) = g(x, x/epsilon) on partial derivative D, where D subset of R-d (d >= 2), is a smooth uniformly convex domain, and g is 1-periodic in its second variable, and both A(epsilon) and g are sufficiently smooth. Our results in this paper are twofold. First we prove L-p convergence results for solutions of the above system and for the non-oscillating operator A(epsilon)(x) = A(x), with the following convergence rate for all 1 <= p < infinity parallel to u(epsilon) - u(0)parallel to (LP(D)) <= C-P {epsilon(1/2p), d = 2, (epsilon vertical bar ln epsilon vertical bar)(1/p), d = 3, epsilon(1/p), d >= 4, which we prove is (generically) sharp for d >= 4. Here u(0) is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8): 1219-1262, 2014), we prove (for certain class of operators and when d >= 3) ||u(epsilon) - u(0)||(Lp(D)) <= C-p[epsilon(ln(1/epsilon))(2)](1/p) for both the oscillating operator and boundary data. For this case, we take A(epsilon) = A(x/epsilon), where A is 1-periodic as well. Some further applications of the method to the homogenization of the Neumann problem with oscillating boundary data are also considered.

QC 20150209

Available from: 2015-02-09 Created: 2015-02-05 Last updated: 2017-06-20Bibliographically approvedOpen this publication in new window or tab >>Equilibrium points of a singular cooperative system with free boundary### Andersson, John

### Shahgholian, Henrik

### Uraltseva, Nina N.

### Weiss, Georg S.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_9_j_idt183_some",{id:"formSmash:j_idt179:9:j_idt183:some",widgetVar:"widget_formSmash_j_idt179_9_j_idt183_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt179_9_j_idt183_otherAuthors",{id:"formSmash:j_idt179:9:j_idt183:otherAuthors",widgetVar:"widget_formSmash_j_idt179_9_j_idt183_otherAuthors",multiple:true}); 2015 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 280, p. 743-771Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Free boundary, Regularity of the singular set, Unique tangent cones, Partial regularity
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-172480 (URN)10.1016/j.aim.2015.04.014 (DOI)000357700800024 ()2-s2.0-84929376546 (Scopus ID)
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##### Funder

Swedish Research Council
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

In this paper we initiate the study of maps minimising the energy integral(D)(vertical bar del u vertical bar(2) + 2 vertical bar u vertical bar) dx, which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations Delta u = u / vertical bar u vertical bar chi({vertical bar u vertical bar >0}), u = (u(1,) ... ,u(m)) Our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps. Our results here concern regularity of the solution as well as that of the free boundary. They are achieved by using monotonicity formulas and epiperimetric inequalities, in combination with geometric analysis.

QC 20150826

Available from: 2015-08-26 Created: 2015-08-25 Last updated: 2017-12-04Bibliographically approved