Concentration inequality

In mathematics, a free boundary problem is a partial differential equation to be solved for both an unknown function u and an unknown domain Ω. The segment Γ of the boundary of Ω which is not known at the outset of the problem is the free boundary.

The classic example is the melting of ice. Given a block of ice, one can solve the heat equation given appropriate initial and boundary conditions to determine its temperature. But, if in any region the temperature is greater than the melting point of ice, this domain will be occupied by liquid water instead. The location of the ice/liquid interface is controlled dynamically by the solution of the PDE.

Two-phase Stefan problems

The melting of ice is a Stefan problem for the temperature field T, which is formulated as follows. Consider a medium occupying a region Ω consisting of two phases, phase 1 which is present when T > 0 and phase 2 which is present when T < 0. Let the two phases have thermal diffusivities α₁ and α₂. For example, the thermal diffusivity of water is 1.4×10⁻⁷ m²/s, while the diffusivity of ice is 1.335×10^{−6 m2/s.}

In the regions consisting solely of one phase, the temperature is determined by the heat equation: in the region T > 0,

$\frac{\partial T}{\partial t}=\mathrm{\nabla }\cdot ({\alpha }_1\mathrm{\nabla }T)+Q$

while in the region T < 0,

$\frac{\partial T}{\partial t}=\mathrm{\nabla }\cdot ({\alpha }_2\mathrm{\nabla }T)+Q.$

This is subject to appropriate conditions on the (known) boundary of Ω; Q represents sources or sinks of heat.

Let Γ_t be the surface where T = 0 at time t; this surface is the interface between the two phases. Let ν denote the unit outward normal vector to the second (solid) phase. The Stefan condition determines the evolution of the surface Γ by giving an equation governing the velocity V of the free surface in the direction ν, specifically

$LV={\alpha }_1{\partial }_{\nu }{T}_1-{\alpha }_2{\partial }_{\nu }{T}_2,$

where L is the latent heat of melting. By T₁ we mean the limit of the gradient as x approaches Γ_t from the region T > 0, and for T₂ we mean the limit of the gradient as x approaches Γ_t from the region T < 0.

In this problem, we know beforehand the whole region Ω but we only know the ice-liquid interface Γ at time t = 0. To solve the Stefan problem we not only have to solve the heat equation in each region, but we must also track the free boundary Γ.

The one-phase Stefan problem corresponds to taking either α₁ or α₂ to be zero; it is a special case of the two-phase problem. In the direction of greater complexity we could also consider problems with an arbitrary number of phases.

Obstacle problems

Another famous free-boundary problem is the obstacle problem, which bears close connections to the classical Poisson equation. The solutions of the differential equation

$-{\mathrm{\nabla }}^{2}u=f,u{|}_{\partial \mathrm{\Omega }}=g$

satisfy a variational principle, that is to say they minimize the functional

$E[u]=\frac{1}{2}\underset{\mathrm{\Omega }}{\int }|\mathrm{\nabla }u{|}^2\mathrm{d}x-\underset{\mathrm{\Omega }}{\int }fu\mathrm{d}x$

over all functions u taking the value g on the boundary. In the obstacle problem, we impose an additional constraint: we minimize the functional E subject to the condition

$u\le \phi$

in Ω, for some given function φ.

Define the coincidence set C as the region where u = φ. Furthermore, define the non-coincidence set N = Ω\C as the region where u is not equal to φ, and the free boundary Γ as the interface between the two. Then u satisfies the free boundary problem

$-{\mathrm{\nabla }}^{2}u=f\text{ in }N,u=g$

on the boundary of Ω, and

$u\le \phi \text{ in }|\mathrm{\Omega },\mathrm{\nabla }u=\mathrm{\nabla }\phi \text{ on }\mathrm{\Gamma }.$

Note that the set of all functions v such that v ≤ φ is convex. Where the Poisson problem corresponds to minimization of a quadratic functional over a linear subspace of functions, the free boundary problem corresponds to minimization over a convex set.

Connection with variational inequalities

Many free boundary problems can profitably be viewed as variational inequalities for the sake of analysis. To illustrate this point, we first turn to the minimization of a function F of n real variables over a convex set C; the minimizer x is characterized by the condition

$\mathrm{\nabla }F(x)\cdot (y-x)\ge 0\text{ for all }y\in C.$

If x is in the interior of C, then the gradient of F must be zero; if x is on the boundary of C, the gradient of F at x must be perpendicular to the boundary.

The same idea applies to the minimization of a differentiable functional F on a convex subset of a Hilbert space, where the gradient is now interpreted as a variational derivative. To concretize this idea, we apply it to the obstacle problem, which can be written as

$\underset{\mathrm{\Omega }}{\int }({\mathrm{\nabla }}^{2}u+f)(v-u)\mathrm{d}x\ge 0\text{ for all }v\le \phi .$

This formulation permits the definition of a weak solution: using integration by parts on the last equation gives that

$\underset{\mathrm{\Omega }}{\int }\mathrm{\nabla }u\cdot \mathrm{\nabla }(v-u)\mathrm{d}x\le \underset{\mathrm{\Omega }}{\int }f(v-u)\mathrm{d}x\text{ for all }v\le \phi .$

This definition only requires that u have one derivative, in much the same way as the weak formulation of elliptic boundary value problems.

Regularity of free boundaries

In the theory of elliptic partial differential equations, one demonstrates the existence of a weak solution of a differential equation with reasonable ease using some functional analysis arguments. However, the weak solution exhibited lies in a space of functions with fewer derivatives than one would desire; for example, for the Poisson problem, we can easily assert that there is a weak solution which is in H¹, but it may not have second derivatives. One then applies some calculus estimates to demonstrate that the weak solution is in fact sufficiently regular.

For free boundary problems, this task is more formidable for two reasons. For one, the solutions often exhibit discontinuous derivatives across the free boundary, while they may be analytic in any neighborhood away from it. Secondly, one must also demonstrate the regularity of the free boundary itself. For example, for the Stefan problem, the free boundary is a C^1/2 surface.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.