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| 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'''.
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| 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.
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| == Two-phase Stefan problems ==
| | Absinthe one of the finest liquor of all times has produced a handsome comeback after being banned for almost a century. Absinthe accessories as well as other barware like absinthe glasses, absinthe spoons and other absinthe items have been in great demand the world over. Part of the demand is due to the history and the culture associated with it. Absinthe is a drink that has to be ready by following a traditional ritual. There are 2 very commonly followed customs to make the absinthe drink.<br><br>Conventional French ritual requires an absinthe glass, absinthe, special absinthe spoon, sugar cubes, and ice cold water. The [http://Answers.yahoo.com/search/search_result?p=initial+step&submit-go=Search+Y!+Answers initial step] involves pouring absinthe dose inside the special absinthe glass which has an etched line to indicate the quantity of absinthe. A special flat perforated absinthe spoon is placed on top of the cup and a sugar cube is put on the spoon. Ice cold water is then dripped over the cube so it dissolves in the dripping drinking wateh2o and falls in the glass containing absinthe. As the absinthe gets watered down it turns from clear green to opaque white, this is known as the louche effect. Louching happens as essential oils from the herbs such as wormwood becomes precipitated. <br><br>Absinthe spoons have a very significant part in the absinthe practice and as such they are an essential accessory in bars and cafés that serve absinthe. The absinthe spoon is punctured and slotted in order to permit the sugar and ice-cold water to seep through. Absinthe has large alcohol content and incorporating water is a necessity. Initially absinthe spoon were only functional however by the end of the 1870s absinthe spoons got a really decorative and refined look. Numerous designs, style, and forms of absinthe spoons were seen at the cafes.<br><br>A few of the styles and designs are artwork and more often their style and form represent a period of time in history. In 1889 the Eiffel tower was inaugurated and to mark this historical occasion, absinthe spoon was developed in the shape of the Eiffel tower. This is the most preferred style of spoon. The first Eiffel tower spoon nowadays directions a premium in the antique industry. Some spoons resembled leaves while some had geometrical designs. Should you have just about any queries about where in addition to tips on how to work with [http://www.Google.com Discover more], you are able to call us with our webpage. Absinthe spoons have been also employed for publicity by producers of absinthe.<br><br>Absinthe spoons still possess a heavy price tag. The commemorative and also antique spoons are collectibles and command a premium. Even other accessories including fountains that have been utilized to pour cold water over sugar cubes, and absinthe glasses are much sought after. Unique absinthe spoons are get more information hard to separate from the knockoffs that are being peddled as originals. One secret tip is to try to find the stamping as original spoons were stamped whilst fakes are molded. Also original spoons will have shown some defects and differences whenever two original absinthe spoons are in contrast.<br><br>In the event you don't want to purchase the authentic absinthe it is possible to opt for exact replicas for as less as $10. There are a number of sites that sell absinthe spoons as well as other absinthe accessories at inexpensive price points. One particular place is absinthekit.com. |
| 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 diffusivity|thermal diffusivities]] α<sub>1</sub> and α<sub>2</sub>. For example, the thermal diffusivity of water is 1.4×10<sup>−7</sup> m<sup>2</sup>/s, while the diffusivity of ice is 1.335×10<sup>−6</sub> m<sup>2</sup>/s.
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| In the regions consisting solely of one phase, the temperature is determined by the heat equation: in the region ''T'' > 0,
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| : <math> \frac{\partial T}{\partial t} = \nabla\cdot(\alpha_1 \nabla T) + Q</math>
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| while in the region ''T'' < 0,
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| : <math> \frac{\partial T}{\partial t} = \nabla\cdot (\alpha_2\nabla T) + Q.</math>
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| This is subject to appropriate conditions on the (known) boundary of Ω; Q represents sources or sinks of heat.
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| Let Γ<sub>t</sub> 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
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| : <math>LV = \alpha_1\partial_\nu T_1 - \alpha_2\partial_\nu T_2,</math>
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| where ''L'' is the latent heat of melting. By ''T''<sub>1</sub> we mean the limit of the gradient as ''x'' approaches Γ<sub>t</sub> from the region ''T'' > 0, and for ''T''<sub>2</sub> we mean the limit of the gradient as ''x'' approaches Γ<sub>t</sub> from the region ''T'' < 0.
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| 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.
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| The one-phase Stefan problem corresponds to taking either α<sub>1</sub> or α<sub>2</sub> 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. | |
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| == Obstacle problems == | |
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| Another famous free-boundary problem is the [[obstacle problem]], which bears close connections to the classical [[Poisson equation]]. The solutions of the differential equation
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| :<math> -\nabla^2 u = f, \qquad u|_{\partial\Omega} = g</math>
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| satisfy a variational principle, that is to say they minimize the functional
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| :<math> E[u] = \frac{1}{2}\int_\Omega|\nabla u|^2 \, \mathrm{d}x - \int_\Omega fu \, \mathrm{d}x</math>
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| 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 | |
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| :<math> u \le \varphi \, </math>
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| in Ω, for some given function φ.
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| 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
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| :<math> -\nabla^2 u = f\text{ in }N,\quad u = g</math>
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| on the boundary of Ω, and
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| :<math> u \le \varphi\text{ in }|\Omega,\quad \nabla u = \nabla\varphi\text{ on }\Gamma. \, </math>
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| 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.
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| == Connection with variational inequalities ==
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| Many free boundary problems can profitably be viewed as [[variational inequality|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
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| :<math>\nabla F(x)\cdot(y-x) \ge 0\text{ for all }y\in C. \, </math>
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| 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.
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| 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
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| :<math>\int_\Omega(\nabla^2 u + f)(v - u) \, \mathrm{d}x \ge 0\text{ for all }v \le \varphi.</math>
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| This formulation permits the definition of a weak solution: using integration by parts on the last equation gives that
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| :<math>\int_\Omega\nabla u\cdot\nabla(v-u)\mathrm{d}x \le \int_\Omega
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| f(v-u) \, \mathrm{d}x\text{ for all } v \le \varphi.</math>
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| This definition only requires that ''u'' have one derivative, in much the same way as the weak formulation of elliptic boundary value problems.
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| == Regularity of free boundaries ==
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| In the theory of [[elliptic operator|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 [[Sobolev space|''H''<sup>1</sup>]], but it may not have second derivatives. One then applies some calculus estimates to demonstrate that the weak solution is in fact sufficiently regular.
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| 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''<sup>1/2</sup> surface.
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| == References ==
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| <!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->
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| *{{citation|last=Alexiades|first=Vasilios|title=Mathematical Modeling of Melting and Freezing Processes|publisher=Hemisphere Publishing Corporation|year=1993|isbn=1-56032-125-3}}
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| *{{citation|last=Friedman|first=Avner|title=Variational Principles and Free Boundary Problems|publisher=John Wiley and Sons, Inc.|year=1982|isbn=978-0-486-47853-1}}
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| *{{citation|last=Kinderlehrer|first=David|last2=Stampacchia|first2=Guido|title=An Introduction to Variational Inequalities and Their Applications|publisher=Academic Press|year=1980|isbn=0-89871-466-4}}
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| {{Reflist}}
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| <!--- Categories --->
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| [[Category:Articles created via the Article Wizard]]
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| [[Category:Partial differential equations]]
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Absinthe one of the finest liquor of all times has produced a handsome comeback after being banned for almost a century. Absinthe accessories as well as other barware like absinthe glasses, absinthe spoons and other absinthe items have been in great demand the world over. Part of the demand is due to the history and the culture associated with it. Absinthe is a drink that has to be ready by following a traditional ritual. There are 2 very commonly followed customs to make the absinthe drink.
Conventional French ritual requires an absinthe glass, absinthe, special absinthe spoon, sugar cubes, and ice cold water. The initial step involves pouring absinthe dose inside the special absinthe glass which has an etched line to indicate the quantity of absinthe. A special flat perforated absinthe spoon is placed on top of the cup and a sugar cube is put on the spoon. Ice cold water is then dripped over the cube so it dissolves in the dripping drinking wateh2o and falls in the glass containing absinthe. As the absinthe gets watered down it turns from clear green to opaque white, this is known as the louche effect. Louching happens as essential oils from the herbs such as wormwood becomes precipitated.
Absinthe spoons have a very significant part in the absinthe practice and as such they are an essential accessory in bars and cafés that serve absinthe. The absinthe spoon is punctured and slotted in order to permit the sugar and ice-cold water to seep through. Absinthe has large alcohol content and incorporating water is a necessity. Initially absinthe spoon were only functional however by the end of the 1870s absinthe spoons got a really decorative and refined look. Numerous designs, style, and forms of absinthe spoons were seen at the cafes.
A few of the styles and designs are artwork and more often their style and form represent a period of time in history. In 1889 the Eiffel tower was inaugurated and to mark this historical occasion, absinthe spoon was developed in the shape of the Eiffel tower. This is the most preferred style of spoon. The first Eiffel tower spoon nowadays directions a premium in the antique industry. Some spoons resembled leaves while some had geometrical designs. Should you have just about any queries about where in addition to tips on how to work with Discover more, you are able to call us with our webpage. Absinthe spoons have been also employed for publicity by producers of absinthe.
Absinthe spoons still possess a heavy price tag. The commemorative and also antique spoons are collectibles and command a premium. Even other accessories including fountains that have been utilized to pour cold water over sugar cubes, and absinthe glasses are much sought after. Unique absinthe spoons are get more information hard to separate from the knockoffs that are being peddled as originals. One secret tip is to try to find the stamping as original spoons were stamped whilst fakes are molded. Also original spoons will have shown some defects and differences whenever two original absinthe spoons are in contrast.
In the event you don't want to purchase the authentic absinthe it is possible to opt for exact replicas for as less as $10. There are a number of sites that sell absinthe spoons as well as other absinthe accessories at inexpensive price points. One particular place is absinthekit.com.