What does Buddhism say that the laity should do when someone injuries them?

The message of Buddha is similar, althought he did not give a direct advice how a lay person should react on harm to lay people but indirect in gaving given absolute valide precepts: not harming and destroing life (incl. ALL living beings, also not for food), not to take what is not given, not to speak what is not true and not to misbehave in fleshes lust and not to take indoxicants, which easy cause such.

As for monks, and people who are intented to liberation, the position is and advice is more direct and clear given, not intended for the world but beyound and most famous is the saw-simily:

{patient endurance} MN 21, MN 28

There will follow a good collection of other sayings in this regard and this clear approach in many good answers.

Other than Jesus appearing and how he wins people, Buddha does not advice at it is heroic to and on the cross in serving and try to help others, he does not advice to dy for others, but for those who desire real benefit and liberation, even to dy for the training in keeping his discipline and told that his monks would rather like to dy then to break the basic and importand precepts.

So there is no evidence of advices for sacrify one self for the world and its purpose, at least to help others in the best possible way one can.

One who seeks for revenge will never be at peace, thats of cource the same message in this regard but martyr in this regard has no place and is not praised by the Buddha and his informed followers, also to act for possible display and transport such. Of course people often preach of how many sacrifies the Buddha did, gave his life, to become a Buddha to push on a certain "feel obligated" botton to win followers, but he himself did not.

Just to try to point out the differnces, not so easy to trace and would require much more explaining to understand.

(Note: This is a gift of Dhamma, not mean for commercial purpose or other wordily gains.)

There are various ways to handle this situation. These instructions are mainly for monks. This matter is discussed in detail in the following Sutta.

As a lay person, you can handle the situation like reporting the matter to the police etc.Perhaps you can stop the person with nonviolence manner like self-defense. These are my personal opinion.

http://www.accesstoinsight.org/lib/study/nonviolence.html

Please see the question on "Can or should a Buddhist defend themselves physically?"

Self-defense is OK. According to Mahaparinibbana Sutta:

"What have you heard, Ananda: do the Vajjis duly protect and guard the arahats, so that those who have not come to the realm yet might do so, and those who have already come might live there in peace?"

"I have heard, Lord, that they do."

"So long, Ananda, as this is the case, the growth of the Vajjis is to be expected, not their decline."

However, we should not bear ill-will or hatred in our minds, even if we're defending ourselves. From the Kakacupama Sutta:

"Monks, even if bandits were to savagely sever you, limb by limb, with a double-handled saw, even then, whoever of you harbors ill will at heart would not be upholding my Teaching. Monks, even in such a situation you should train yourselves thus: 'Neither shall our minds be affected by this, nor for this matter shall we give vent to evil words, but we shall remain full of concern and pity, with a mind of love, and we shall not give in to hatred. On the contrary, we shall live projecting thoughts of universal love to those very persons, making them as well as the whole world the object of our thoughts of universal love — thoughts that have grown great, exalted and measureless. We shall dwell radiating these thoughts which are void of hostility and ill will.' It is in this way, monks, that you should train yourselves.

If someone insults you and you don't take it in personally, then it "goes back" to that person. Similarly, any kind of ill-will from others should not be taken personally. From the Akkosa Sutta:

"And what do you think: Do you serve them with staple & non-staple foods & delicacies?"

"Yes, sometimes I serve them with staple & non-staple foods & delicacies."

"And if they don't accept them, to whom do those foods belong?"

"If they don't accept them, Master Gotama, those foods are all mine."

"In the same way, brahman, that with which you have insulted me, who is not insulting; that with which you have taunted me, who is not taunting; that with which you have berated me, who is not berating: that I don't accept from you. It's all yours, brahman. It's all yours.

"Whoever returns insult to one who is insulting, returns taunts to one who is taunting, returns a berating to one who is berating, is said to be eating together, sharing company, with that person. But I am neither eating together nor sharing your company, brahman. It's all yours. It's all yours."

It is up to each situation. Some case running away is proper, but some case negotiating is proper, etc.

The theory is "The buddhist people will try to keep 1)own benefit, 2)other benefits, 3)public benefit, 4)this life benefit, 5)next life benefit, and 6)nibbāna benefit. They try to keep whole 6 benefits, no bias."

Methods, to keep benefits, are multiple up to consciousness management ability of each person.