Quadrature Domains and Equilibrium on the Sphere Alan Legg Department of Mathematical Sciences, Purdue Fort Wayne MWAA October 7, 2018 Alan Legg Quadrature Domains and Equilibrium on the Sphere
This talk will represent work with P. Dragnev, featuring helpful input from E. Saff, building on a paper of Brauchart, Dragnev, Saff, Womersley (2018) (more on that later). Alan Legg Quadrature Domains and Equilibrium on the Sphere
This talk will represent work with P. Dragnev, featuring helpful input from E. Saff, building on a paper of Brauchart, Dragnev, Saff, Womersley (2018) (more on that later). Our interest will be in trying to use the sphere S 2 as the setting for a problem familiar from 2D potential theory: Alan Legg Quadrature Domains and Equilibrium on the Sphere
This talk will represent work with P. Dragnev, featuring helpful input from E. Saff, building on a paper of Brauchart, Dragnev, Saff, Womersley (2018) (more on that later). Our interest will be in trying to use the sphere S 2 as the setting for a problem familiar from 2D potential theory: How does a charge placed on a conductor distribute to obtain a configuration of minimal energy, in the presence of an electric field? Alan Legg Quadrature Domains and Equilibrium on the Sphere
This talk will represent work with P. Dragnev, featuring helpful input from E. Saff, building on a paper of Brauchart, Dragnev, Saff, Womersley (2018) (more on that later). Our interest will be in trying to use the sphere S 2 as the setting for a problem familiar from 2D potential theory: How does a charge placed on a conductor distribute to obtain a configuration of minimal energy, in the presence of an electric field? In the plane, with logarithmic interactions between charges, if a charge is placed onto a domain Ω ⊂ C , what is the equilibrium distribution of the charge? Alan Legg Quadrature Domains and Equilibrium on the Sphere
For example, on a bounded smooth finitely connected domain Ω, we expect the charge to repulse itself as far as possible and reside only on the outer boundary. Alan Legg Quadrature Domains and Equilibrium on the Sphere
In mathematical terms, given a positive unit Borel measure µ as a charge distribution, say with compact support in ¯ Ω, its potential at point z in the plane is: Alan Legg Quadrature Domains and Equilibrium on the Sphere
In mathematical terms, given a positive unit Borel measure µ as a charge distribution, say with compact support in ¯ Ω, its potential at point z in the plane is: 1 � U µ ( z ) = ln | z − w | d µ ( w ) . Alan Legg Quadrature Domains and Equilibrium on the Sphere
In mathematical terms, given a positive unit Borel measure µ as a charge distribution, say with compact support in ¯ Ω, its potential at point z in the plane is: 1 � U µ ( z ) = ln | z − w | d µ ( w ) . The logarithmic energy of the charge distribution is: Alan Legg Quadrature Domains and Equilibrium on the Sphere
In mathematical terms, given a positive unit Borel measure µ as a charge distribution, say with compact support in ¯ Ω, its potential at point z in the plane is: 1 � U µ ( z ) = ln | z − w | d µ ( w ) . The logarithmic energy of the charge distribution is: 1 � � � U µ d µ = I µ = ln | z − w | d µ ( w ) d µ ( z ) . Alan Legg Quadrature Domains and Equilibrium on the Sphere
In mathematical terms, given a positive unit Borel measure µ as a charge distribution, say with compact support in ¯ Ω, its potential at point z in the plane is: 1 � U µ ( z ) = ln | z − w | d µ ( w ) . The logarithmic energy of the charge distribution is: 1 � � � U µ d µ = I µ = ln | z − w | d µ ( w ) d µ ( z ) . The minimal energy problem is to determine which positive unit Borel measure µ supported in ¯ Ω will minimize I µ . Alan Legg Quadrature Domains and Equilibrium on the Sphere
Potential Theory can be used to prove that in normal circumstances (when the ‘conductor’ Ω has positive logarithmic capacity): Alan Legg Quadrature Domains and Equilibrium on the Sphere
Potential Theory can be used to prove that in normal circumstances (when the ‘conductor’ Ω has positive logarithmic capacity): The minimal energy inf µ I µ is finite and obtained by some µ . Alan Legg Quadrature Domains and Equilibrium on the Sphere
Potential Theory can be used to prove that in normal circumstances (when the ‘conductor’ Ω has positive logarithmic capacity): The minimal energy inf µ I µ is finite and obtained by some µ . The energy-minimizing measure is unique. Call it µ E , the equilibrium measure. Alan Legg Quadrature Domains and Equilibrium on the Sphere
Potential Theory can be used to prove that in normal circumstances (when the ‘conductor’ Ω has positive logarithmic capacity): The minimal energy inf µ I µ is finite and obtained by some µ . The energy-minimizing measure is unique. Call it µ E , the equilibrium measure. In fact, for a bounded finitely connected conductor, the equilibrium measure resides only on the outer boundary, as expected. Alan Legg Quadrature Domains and Equilibrium on the Sphere
Potential Theory can be used to prove that in normal circumstances (when the ‘conductor’ Ω has positive logarithmic capacity): The minimal energy inf µ I µ is finite and obtained by some µ . The energy-minimizing measure is unique. Call it µ E , the equilibrium measure. In fact, for a bounded finitely connected conductor, the equilibrium measure resides only on the outer boundary, as expected. There are general results about how to identify the equilibrium measure, and how to identify its particular properties. Alan Legg Quadrature Domains and Equilibrium on the Sphere
What happens in the presence of an external electric field Q ? Alan Legg Quadrature Domains and Equilibrium on the Sphere
What happens in the presence of an external electric field Q ? We expect that the external field will push on the charge distribution, and perhaps deform it. Alan Legg Quadrature Domains and Equilibrium on the Sphere
What happens in the presence of an external electric field Q ? We expect that the external field will push on the charge distribution, and perhaps deform it. In general, the possible equilibrium supports in this case are much more diverse. Alan Legg Quadrature Domains and Equilibrium on the Sphere
What happens in the presence of an external electric field Q ? We expect that the external field will push on the charge distribution, and perhaps deform it. In general, the possible equilibrium supports in this case are much more diverse. This problem can also be analyzed in the plane with logarithmic potential theory, notably as presented by Saff and Totik in their book on the subject. Alan Legg Quadrature Domains and Equilibrium on the Sphere
For the right kinds of external fields (called ‘admissible’), there is a theory of equilibrium measures which has similarities to the theory without external fields. Alan Legg Quadrature Domains and Equilibrium on the Sphere
For the right kinds of external fields (called ‘admissible’), there is a theory of equilibrium measures which has similarities to the theory without external fields. For a conductor Ω in the plane in the presence of an external field Q , place a unit charge on Ω according to a charge distribution µ . The external field influences the energy of the system. Alan Legg Quadrature Domains and Equilibrium on the Sphere
For the right kinds of external fields (called ‘admissible’), there is a theory of equilibrium measures which has similarities to the theory without external fields. For a conductor Ω in the plane in the presence of an external field Q , place a unit charge on Ω according to a charge distribution µ . The external field influences the energy of the system. The weighted energy of the measure µ in the presence of Q is: Alan Legg Quadrature Domains and Equilibrium on the Sphere
For the right kinds of external fields (called ‘admissible’), there is a theory of equilibrium measures which has similarities to the theory without external fields. For a conductor Ω in the plane in the presence of an external field Q , place a unit charge on Ω according to a charge distribution µ . The external field influences the energy of the system. The weighted energy of the measure µ in the presence of Q is: � � 1 � I Q µ = ln | z − w | d µ ( z ) d µ ( w ) + 2 Q ( z ) d µ ( z ) . Alan Legg Quadrature Domains and Equilibrium on the Sphere
For the right kinds of external fields (called ‘admissible’), there is a theory of equilibrium measures which has similarities to the theory without external fields. For a conductor Ω in the plane in the presence of an external field Q , place a unit charge on Ω according to a charge distribution µ . The external field influences the energy of the system. The weighted energy of the measure µ in the presence of Q is: � � 1 � I Q µ = ln | z − w | d µ ( z ) d µ ( w ) + 2 Q ( z ) d µ ( z ) . The equilibrium measure in the presence of Q is then the positive unit Borel measure which will minimize the weighted energy I Q µ . Alan Legg Quadrature Domains and Equilibrium on the Sphere
A fundamental tool in this situation is the following Frostman theorem: Alan Legg Quadrature Domains and Equilibrium on the Sphere
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